One minor phonological innovation in Finnish is mentioned in historical overviews far more often than could be expected from its lexical frequency: the loss of a palatal semivowel *j when preceding its vocalic counterpart *i. This is probably because the shift has been fossilized as a morphological alternation  in the word veli ‘brother’ (< *velji), stem velje-. The change also shows up in some old derivatives, e.g. nelikko ‘group of four’ (< *neljikko) from neljä ‘four’.
For phonological analysis, both synchronic and diachronic, a principle that I find valuable is back/front symmetry. This follows as a special case of what is perhaps the main result of featural phonology: phonemes are not atomic entities, but rather bundles of features. And so sound changes or phonological processes that are conditioned on vowel height tend to ignore vowel backness and roundedness. Here we would then expect to also find the corresponding shift involving labial (semi)vowels: pre-Finnic *-w- or proto-Finnic *-v- > ∅ before *u or *ü (= in shorthand: *U).
Yet it turns out that this question is barely discussed anywhere. I have e.g. found no mention of such a development in Lauri Hakulinen’s Suomen kielen rakenne ja kehitys.  Martti Rapola’s Suomen kielen äännehistorian luennot does not fare much better (as in perhaps predictable though, since his focus is firmly on dialectal developments within Finnish, not on pan-Finnish innovations).
Let’s try having a look if there is any evidence to be found on this matter.
Given the absense of clear evidence for *U-stems in Proto-Uralic times, there are not many words where we can reasonably assume the sequence *-wU- to have existed in pre-Finnic times. Just one clear word-initial case of loss can be found: *wülä- > PF *ülä- ‘up(per)’ — cf. Permic *vɨl-.  Slightly odder is *wud₂ə ‘new’ (and even this, I believe, should be regardless derived from an even earlier *wod₂ə, though this is of no direct relevance for the current topic). This turns up as PF *uuci (Fi. uusi etc.), seemingly with vocalization, rather than loss, of the initial glide. We could also e.g. assume a metathesis *wu- > *uw- as an intermediate stage.
Still, Proto-Finnic clearly had *u-stems, whatever their origin. And it seems that there is still a decent amount of of evidence for a simplification *-wU- > *-U- in these. Already within Finnish I can find three clear doublets involving word derivation:
- kalvaa ‘to gnaw’ ~ kaluta ‘id.’ (< ? *kalvuta) 
- kärventää ‘to scorch’ ~ käry ‘burnt smell, rancor’ (< ? *kärvü)
- raivo ‘fury’ ~ raju ‘fierce’ (< ? *raivu)
Comparison with Samic also turns up three likely cases.
- Lule Sami iellvet ‘to note’ (< ? PS *ealvē-) ~ Fi. äly ‘intellect’, älytä ‘to realize’ (? < *älv-ü)
- Proto-Samic *ocvē ‘wet snow’ (< *učwa) ~ Fi. utu ‘mist, fog’ (< ? *učw-u) 
- Proto-Samic *toalvō- ‘to lead, to take somewhere’ (< *tolvo- < ? *talwəw-) ~ Fi. taluttaa ‘to lead, to walk someone’ (< ? *talvu-tta- < *talwəw-)
I hypothesize that a close scan of *U-stem roots and derivatives in the other Finnic languages would turn up further evidence as well.
Much like is the case with -ji-, Modern Finnish does however allow the sequence -vU-.
Many of these cases can be shown to have been formed secondarily, and could be hypothesized to have come about only after *-v-loss. E.g. some go back to earlier *-βu- < *-bu- (I give here only non-paradigmatically-alternating cases):
- juovu-ttaa < *joobu-tta- ‘to get/make someone drunk’ (← juopua ‘to become drunk’)
- taivu-ttaa < *taibu-tta- ‘to bend’ (← taipua ‘to bend’)
- vaivu-ttaa < *vaibu-tta- ‘to sink (tr.), lull’ (← vaipua ‘to sink, to fall asleep’)
- viivy-ttää < *viibü-ttä- ‘to delay’ (← viipyä ‘to be late’)
- voivu-ttaa < *voibu-tta- ‘to tire (tr.)’ (← voipua ‘to tire (intr.)’)
some involve loaning:
- laavu ‘lean-to’ ← Samic, cf. e.g. NS lávvu ‘id.’
- siivu ‘slice’ ← Swedish skiv ‘id.’
- laiv-uri ‘skipper’ (← laiva ‘ship’; -Uri is a loan suffix from Swedish)
- päiv-yri ‘almanac’ (← päivä ‘day’)
and others yet result from a late assimilation of unstressed *-AU- to -UU-: 
- arv-uuttaa < *arvautta- < *arvad-u-tta ‘to ask riddles’ (← arvata ‘ to guess’)
- raiv-uu < *raivau < *raivad-u ‘clearing’ (← raivata ‘to clear land, etc.’)
- tavu ‘syllable’ < older †tavuu < *tavau < *tavad-u (← tavata ‘to spell’)
A few remaining derivative examples could be assumed to have been formed only after *-v-loss, or to have been reverted by analogy.
- harv-uus < *harv-us ‘sparseness’ (← harva ‘sparse’) 
- kaiv-u ‘digging, trench’ (← kaivaa ‘to dig’; this is an IMO unetymological doublet of *kajwa-w > kaivo ‘well’)
- kasv-u ‘growth’ (← kasvaa ‘to grow’; the phonologically expected kasvo already means ‘face’)
- kuiv-u- ‘to dry’ (← kuiva ‘dry’)
A soundlawful  doublet of the last one is possibly found in dialectal kujua ‘to wilt’.
Regardless, there remains a more problematic residue, which prevents me from simply assuming that *-vU- always > *-U- at some relatively early Finnic period. These are all basic noun roots with primary *-v-, where morphophonological alternation as a source of analogy cannot be possibly blamed for anything.
- koivu ‘birch’. The only real excuse I could think up here is that in South Estonian the root is instead an o-stem, kõiv : kõivo-. So perhaps there has been here a later shift from *-vo to *-vu in North Finnic…? (The root has not been attested from North Estonian; in Votic it probably only occurs as an Ingrian loan; Livonian provides no evidence for the distinction between *-o and *-u.) This would still not be a regular sound change though, given aivo ‘brain’, arvo ‘value’, hieho < *hehvo ‘heifer’, kalvo ‘film, membrane’, etc. 
- savu ‘smoke’ seems like it might actually be a positive example of the change, to an extent. On the basis of South Estonian sau ~ Votic and dialectal Olonets Karelian savvu  it would be possible to reconstruct PF *savvu; then, just as could be predicted, one *-v- is lost in Finnish. However, this only leads to the question: why does *-v-loss not occur in the previous three varieties as well? Its loss is still seen in e.g. ‘mist’: SEs udsu, NEs udu, Votic utu.
An explanation may lie in the earlier history of this word. Samic *sōvë ‘smoke’ and Mordvinic *suf-ta- ‘to smoke’ indicate that the earlier form of the root was simply *sawə, not anything like **sawəw. Erkki Itkonen has supposed  that the Finnish word is not formed by suffixation, but rather by apocope-then-anaptyxis. In PF times, all former bisyllabic words ending in *-jə were contracted into diphthongs (e.g. *täjə > *täi ‘tick’, *wajə > *woojə > *voi ‘butter’); so in parallel, we would then expect also *sawə to have been contracted to *sau. But no nominal roots of the shape ˣCVU occur in the native lexicon of Finnish (and the scarce loanwords such as tau ‘tau’ or tiu ’20 items’ are on the recent side as well). Itkonen therefore posits a back-development *sau > savu, to better abide with the canonical bisyllabic root structure. The South Estonian form could then be considered an archaism. Perhaps likewise also the identical monosyllabic reflexes in Southwestern Finnish; although since SW Finnish clearly has had contraction in secondary cases with *-Vbu- > -Vvu- > -Vu- (papu ‘bean’ : SW plural pau ~ standard pavut), this wouldn’t really provide any additional sound change economy.
- vävy ‘son-in-law’ is almost entirely parallel to the above. We again have North Estonian väi, South Estonian väü, Olonetsian vävvy, suggesting PF *vävvü — although, this time Votic shows shorter vävü. We could well again follow Itkonen’s solution and assume PF *väü. On the other hand, Samoyedic *weŋü suggests to me that the proto-form could this time have been something like *wEŋəwə, predicting indeed PF *vävü < *wäwəw. 
- havu ‘conifer branch’. This could again come from *hau > *havvu > havu, as per Itkonen, in light of Olonetsian havvu. On the other hand, a loan etymology from Baltic (cf. Lithuanian žabas ‘branch’) and Ludian/Veps habu suggest that the proto-form was actually *hapu (with exceptional widespread levelling to the weak-grade stem), or perhaps *habu (with an exceptional unalternating *b).
- sivu ‘side’. This word definitely does not seem to go back to **sivvu / **siu, given Olonetsian sivu. It might be possible to derive this as a Germanic loanword, in which case this could again be analyzed as a late-comer, but there are several phonological difficulties (e.g. what Old Norse actually has is síða< *sīdǭ, not the seemingly required ˣsíð < **sīdu < **sīdō; western Finnish dialects do not have forms along the lines of ˣsiru or ˣsilu that would be predicted from earlier *siðu; vowel length would be expected to remain in a sufficiently recent loan).
This leads me to suggest that the shift *-vU- > -U- has only taken place following another consonant. Most of my six initial examples are compatible with this. In case of koivu, we’d need to assume this got its -u only after the phonologization of *-oj as the diphthong /oi/; while raju and kujua might need to be analyzed as having originated in western Finnish specifically and spread from there to other varieties. Itkonen’s account of savu and vävy continues to work too, since the key forms like savvu show a geminate -vv-, not a diphthong + glide ˣsauvu (as modern Finnish prefers in cases like this, e.g. sauva ‘pole’). But we could also take a slight shortcut, supposing that these never had a geminate in most of Finnic, and that -vv- in Olonetsian (and Votic?) is indeed a late local innovation rather than an archaism.
In one broad stroke, this conditioning also takes care of just about all of the counterexamples above that could perhaps involve secondary counterfeeding (the types of juovuttaa, laavu, raivuu, kaivu). Additionally, among the positive examples, in one case the involved -v- might indeed derive earlier *-b-: kärventää ‘to scorch’ (tr.) seems like an affective/ideophonic variant of korventaa ‘id.’, which is derived from korveta (: korpeaa) ‘to scorch’ (intr.) < PU *korpə-.
As a third line of evidence in favor of this approach, let’s note that *-ji- > *-i- also seems to not take place following a vowel (laji ‘kind, species’, lujin ‘hardest’ ← luja ‘hard’, nuijia ‘to clobber’ ← nuija ‘club’, ojittaa ‘to dig ditches’ ← oja ‘ditch’) and is probably a post-Proto-Finnic change (*velji ‘brother’ > Karelian veľľi ~ velli, Votic velli). Maybe even particular to Finnish! Es. veli can be derived just as well through apocopated *velj (compare e.g. *neljä > *nelj > neli ‘4’).
Tracing the implications further, I even suspect that cases like PU *täjə > PF *täi = Fi. täi ‘tick’; PU *wajə > *woojə > PF *voi = Fi. voi, as mentioned above, have probably not develeped through a stage such as *täji, *vooji — but have involved the direct apocope of PU *-ə following a glide. In principle this predicts that words of the shape *CVji would perhaps have been possible already by Proto-Northern Finnic, from PF *CVjei < earlier *CVjA-j. Suitable roots for forming derivatives of this kind were rare, though.
This may seem to create problems for accounting for words of the shape CVvi : CVve-, like PF *kivi = Fi. Es. etc. kivi ‘stone’… but by now I have, also for other reasons, ended up with the hypothesis that these involve either the levelling of earlier alternation (*kiü : *kive- → *kivi : *kive-), or a geminate in Proto-Finnic that blocked this apocope (e.g. *povvi ‘bosom’ > Fi. povi, Votic põvvi, Es. *põvv > põu).
A second group — and more?
I have not exhausted above the examples known to me where a development *-vU- > -U- could be supposed for Finnish (or elsewhere in Finnic). However, all words remaining up my sleeve show some ambiguity: they involve syllable contraction *-VvU- > -VU-, and they could be analyzed also as cases of syncope followed by vocalization: *-VvU(C…) > *-Vv(C…) > -VU(C…)-. This hypothesis gains some support also from that several examples could have involved the loss of some vowel other than close rounded *-u- or *-ü-. They also commonly enough involve secondary *-v- from *-b-.
The following clearly have involved earlier *-vU-:
- haukka ‘hawk’ < havukka (attested in eastern Fi.!) < *habukka — cf. Veps habuk
- hius (single) hair’ < *hivus < *hibus — cf. Karelian hivus, Veps hibus
- säyseä ‘tame’ < ? *sävüseä — cf. sävyisä ‘id.’; sävy ‘tone, hue’
The following may have had *-vU-, but other possibilities are reasonable as well:
- auttaa ‘to help’ < ? *avu-ttaa / *avi-ttaa; aulis ‘willing to help’ < ? *avu-lis
— cf. apu ‘help’, Veps abutada ‘to help’; or Western Fi. avittaa ‘to help’ (with counterparts in southern Finnic such as Es. aitama)
- keuhko ‘lung’ < ? *kevu-hko / *keve-hko; köykäinen < köyhkäinen ‘light, feeble’ < ? *kevü-hkäinen / *keve-hkäinen
— cf. kevyt ‘light’; or kepeä ‘light’
- liukas ‘slippery’ < ? *livu-kas / *live-kas — cf. lipu ‘slipperyness’; or livetä ‘to slip’, lipeä ‘lye’ (liueta : liukenee ‘to dissolve’, pro ˣlipVeta, and liukua ‘to slide’ have to be analogical; the latter’s soundlawful doublet seems to be lipua ‘to glide’)
- soukka ‘narrow’ < ? *sovu-kka / *sovi-kka — cf. sopukka ‘nook’; or sopia ‘to fit’
The following have no evidence specifically in favor of *-vU-:
- aukko ‘hole’ < ? *ava-kko — cf. avata ‘to open’ (or < ? *auɣekko, cf. auki ‘open’, aueta : aukenee ‘to open’ (intr.); unlikely though given Livonian ouk)
- kiukku ‘anger’ < ? *kiiva-kku — cf. kiivas ‘quick-tempered’
- loukko ‘nook’ < ? *love-kko — cf. lovi : love- ‘cleft’
- reuhtoa ‘to yank around’ < ? *revihtoa / *revehtoa — cf. repiä ‘to tear’ (tr.); revetä ‘to tear’ (intr.)
- riuska ‘brisk’ < ? *rive-ska / *riva-ska — cf. ripeä ‘id.’, rivakka ‘id.’
- saukko ‘otter’ < ? *sava-kkoi — cf. sapa ‘tail’ (but alternately from *sagukkoi, cf. *sagarma(s) ‘otter’ > Es. saarmas, Veps sagarm)
- tiukka ‘tight’ < ? *tiivi-kka — cf. tiivis ‘compact’
- tyyssija ‘abode’ < ? *tyve-s- — cf. tyvi : tyve- ‘base’ (even -yy- < *-yi- might be possible!)
General syncope after -v- however clearly cannot be assumed. Some examples that do not alternate with related bisyllabic forms, even through derivation, include: havista ‘to swish’, havitella ‘to strive for’, hävitä ‘to disappear, lose’, kavala ‘treacherous’, kivahtaa ‘to snap at’, kuvottaa ‘to be/make nauseous’, navakka ‘strong (of wind)’, ovela ‘shrewd’, ravistaa ‘to shake’, ravita ‘to nourish’, sivellä ‘to brush (paint etc.)’, suvanto ‘river pool’. To these could be also added an abundance of more or less transparent derivatives such as avuton ‘helpless’, kivittää ‘to stone’, kovasin ‘whetstone’, lävitse ‘thru’, savinen ‘clay-y’, syventää ‘to deepen’, tavallinen ‘normal’, toivomus ‘wish’, but I believe the point is made without going for completeness.
I could still see some patterns in favor of reconstructing at least conditional syncope. Most of the contracted examples involve following *-kk-; most involve a short first syllable (contrast the juovuttaa ja laavu types earlier); most seem to be “weak grade” formations, where the 2nd syllable would originally have been always closed (including also hius : hiukse-).
But what this is also reminding me of is the pattern of modern colloquial Finnish “clipped” or “slang” derivatives. These are not formed by agglutination, but instead by taking the initial CV(V)C or CVCC sequence of a word, shortening a long vowel if necessary , and appending a suffix after that. Some examples of derivation of this kind include:
- -(t)sa: kot
italous→ kotsa ‘home economics (as a school subject)’ maant ieto→ mantsa ‘geography (as a school subject)’
- -(t)si(-): fund
eerata→ funtsia ‘to think’, kan nattaa→ kantsia ‘to be worth doing’, mil joona→ miltsi ‘million’ (of money), par veke→ partsi ‘balcony’
- -(t)su: fant
astinen→ fantsu ‘fantastic’, rant a→ rantsu ‘beach’; common in nicknames, e.g. An na, An ni, An nika(etc.) → Antsu, Mil la→ Miltsu, Valt teri→ Valtsu
- -(t)ska: jut
tu→ jutska ‘thing(y)’, tiet okone→ tietska ‘computer’
- -(t)ski: jäät
elö→ jätski ‘ice cream’, nuot io→ notski ‘campfire, bonfire’
- –(t)sku: ban
aani→ bansku ‘banana’, mat eriaali→ matsku ‘(reading) material’
And -kka is one of the more productive suffixes of this kind. E.g.
joitus→ harkka ‘training’
le→ junkka ‘jungle’ (the electronic music subgenre!)
a-auto→ linikka ‘bus’
unta→ liikka ‘physical exercise (as a school subject)’
steri‘Master (degree)’ → maikka ‘teacher’
ukumi→ purkka ‘chewing gum’
näs→ Sörkka ~ Sörkkä ‘district in Helsinki’
We also know some examples of this exact derivation pattern whose spread of cognates suggests fairly great age. Three good examples are the informal family terms eukko ‘woman, wife’ (< *emkko?) (cognate in Karelian), probably from emo / emä ‘mother’; ukko ‘man, husband’ (cognates in almost all Finnic languages), from uros ‘male’; veikka, veikko ‘brother, comrade’ (cognates in all Northern Finnic languages), from veli ‘brother’ (< *velji, as mentioned). I take it as probable that clipped derivation has been around for a good millennium or two in Finnic by now, even if it has never been very likely to leave lasting records.
As for examples that could bridge this handful of ancient-looking examples with 20th-century slang, I’m foremost thinking of examples of adjectives showing “suffix alternation”. At least formally, the possibility of reanalysing a stem and agglutinating -kka to that is possible. But nothing really precludes a “clipping” analysis either. E.g.:
- jämeä ‘stiff’ ~ jämäkkä ‘sturdy’ (PU *jämä)
- kimeä ‘high-pitched’ ~ kimakka ‘id.‘ (*kima, √kima?)
- kalpea ‘pale’, kalvas ‘id.‘ ~ kalvakka ‘paleish’ (*kalpa, √kalpa?)
— But even if some of the examples above are indeed clipped derivatives (I would suggest kiukku and tiukka as probable cases, due to e.g. their proto-forms with long vowels), this is unlikely to be the full story either. In particular haukka is not a derivative of any kind, but rather a loan in its entirety (← Proto-Germanic *habukaz).
Since it seems futile to cover the remaining cases by a single rule, it is probably wise to not attempt this. I am therefore leaning towards the option that there are no less than three similar but distinct sound changes involved here:
- *V̆vU > VU, in western Finnish (the haukka and also pau, koju type)
- *CvU > CU, across all Finnish varieties, perhaps most of Finnic, though later than *b > *β > v (the käry, taluttaa type)
- *Vwə > *VU, in Proto-Finnic times under so far unclear conditions (a few e-stem derivatives such as loukko and tyys-; possibly the savu group).
Type 3 seems moreover likely to be identical to the rise of some Proto-Finnic instances of long *UU: e.g. PU *śowə > WU *śuwə > *śuw > PF *suu = Fi. Es. etc. suu ‘mouth’; PU *tiwənə > *tiwnə > *tiüni > PF *tüüni = Fi. tyyni ‘calm’. 
It remains to be seen how well an analysis of data also from outside Finnish will support this division. To reiterate, I would in particular predict being able to find some further examples of type 2 from the other Finnic languages, involving derivatives in -U that have no exact Finnish counterparts.
An initial blind test already turns up at least one candidate in confirmation. Taking at random one Finnic root of suitable shape: *harva ‘rare, sparse’, I could predict that a derivative *harv-u would later yield haru. A word of this shape indeed turns out to be attested from southern Karelian, in the reasonably suitable meaning ‘watered-down milk’. But a fuller derivative hunt will have to wait for later.
 I was going to say “morphophonological”, but really my view is that at least some 80% of all “processes” proposed by morphophonologists educated in generative phonology are not synchronic rules of phonology at all, but merely the still-visible historical residue of former diachronic sound changes. In this particular case, too, it’d take far more mental gymnastics or morphophonological epicycles to explain why underlying /velji/ would surface as [ˈʋeli], while e.g. in the plural genitive, apparent underlying /velj-i-en/ surfaces as [ˈʋeljien] — than to simply assume that the nominative of ‘brother’ is stored as the separate lexeme /veli/.
(To be fair, I’ve seen recent generativist work taking the stance that a level of “lexical” phonology between “deep structure” and surface realization needs to be posited after all, e.g. Kiparsky, “Formal and Empirical Issues in Phonological Typology“. This will likely go a good way towards rectifying the situation, but it may still be a while before people will be willing to consider e.g. that most allomorphy can be modelled as simply a subtype of synonymy.)
 So far, anyway. Any book that has ~1200 footnotes will contain much information that is not in the expected place.
 Even here I am actually not fully sure that breaking *ü- > *vɨ- can be ruled out (similar to Mordvinic, where *ü- > *ve-). Reconstructing instead Finno-Permic *ülä- would make it slightly easier to reconcile this with East Uralic *ilə- (> Mansi *äl-, Khanty *eeL-, Samoyedic *i-). But the zero onset in the latter could perhaps also be explained as analogy from *ëla- ‘down’.
 The case of kaluta is mentioned by Rapola; he however entertains also the possibility that they would not involve suffixation, but rather a “Sievertian” development -lv- > -lu- (and, presumably, the resulting trisyllabic stem *kalua- being then reanalyzed as if it were an original contraction stem *kaluda-, hence the modern infinitive kaluta and not kaluaa). There are no exact parallels for such a change; southwestern Finnish has the relatively similar -sv- > -su- (kasuaa ‘to grow’, rasua ‘fat’), but kaluta is pan-Finnish.
 A comparison I have previously proposed in the comments section.
 It would be an interesting question how these derivational cases diverged from *-Abi > *-AU > *-AA in 3rd person singular forms (as in *aja-bi > *ajau > ajaa ‘drives’), but I would presume some analogy in some direction is involved.
 Vowel length in this suffix is, per the usual explanations, due to complicated multi-stage analogy.
 To coin a translation for the useful concept expressed by German lautgesetzlich / Finnish äännelaillinen.
 In southwestern Finnish dialects different forms, such as koju ‘birch’ or aju ‘brain’, can also be found. Influence from Estonian is very much not ruled out though.
 Karjalan kielen sanakirja lists the forms savvu, vävvy and havvu from the southernmost dialects of Olonetsian, in the villages of Kotkatjärvi, Nekkula and Riipuskala.
 Itkonen, Erkki. “Beiträge zur Geschichte der einsilbigen Wortstämme im Finnischen”. — Finnisch-Ugrische Forschungen 30: 1–54.
 With Lehtinen’s Law blocked by the third-mora element, hence not *veevü. — Samic *vīvë is very difficult to account for. The apparent development *-ŋ- > *-v- has previously inspired suggestions of loaning from early Finnic, but in light of also the stem vowel mismatch, something like *wäŋəwə > *weŋəwə > *weŋwə > *wēɣwə > *vējvë > *vīvë (where the original *-ŋ- isn’t what yields *-v-) could also be within the realms of possibility.
 Modern Finnish still disallows overheavy syllables containing a long vowel and a coda cluster. Pointti ‘(rhetorical or score) point’ and jointti ‘marijuana joint’ are possibly first heralds of the syllable structure CVVCC making a more general entrance, but e.g. tietska is rather syllabifiable as tiet.ska, with a word-internal onset cluster, much like we need to assume also for loanwords such as ekstra (= probably eks.tra).
 My account of *üü in here is tentative — it would have to pre-date *ti > *ci, and it’s possible that there are grounds to exclude this ordering. I’ll have to fiddle with my poset model of Proto-Finnic relative chronology to see if this can be made fit in…