Historical/comparative phonology of the Finnic languages has reached remarkably thorough coverage already in the mid-20th century. Nearly all major varieties and numerous smaller dialect groups (particularly but not only of Finnish) have had their specific history covered by at least a large article-sized special study, often indeed a monograph. Where there remains more to do, the issue is mostly of patches such as working out relative chronologies, pathways and areal patterns of change, or proto-forms of specific items.
There is however one case where a full rewrite would be warranted: Ludian, treated for ages as a “mixed Karelian–Veps variety”, but recently finally argued in detail by Miikul Pahomov (2017): Lyydiläiskysymys to be in essence instead a more conservative sibling of Veps. Or more strictly speaking, a cluster of such dialects: there seem to be no exclusively Ludian innovations that could be used to define this as a single language to the exclusion of Veps! The definition of “Ludian” has always been by a specific combination of retentions. E.g. going by the most immediately obvious phonological traits, Ludian retains Proto-Finnic *b *d *g as such (per older views: fortites *β *ð *ɣ to stops) (shared with Veps), but also retains long close vowels (shared with Northern Veps and all the rest of Finnic) and diphthongizes rather than shortens long non-close vowels (ie üö uo < *ee *öö *oo shared widely, iä ua < *ää *aa shared with Karelian and Eastern Finnish). Diphthongization per se remains an innovation here, but it’s too trivially areal to be worth anything for subgrouping. In fact a few sub-dialects even retain ää aa (so do again a few dialects of Karelian and Eastern Finnish), and one of the more poorly documented varieties appears to shorten them, as Veps does. Worth mentioning is also that even the speakers of what is usually called “Northern Veps”, and some of central Veps, in fact call themselves Ludians rather than Vepsians.
So the old two-part monograph on the historical phonology of Ludian by Aimo Turunen — Lyydiläismurteiden äännehistoria I (1946) on consonantism, II (1950) on vocalism — seems to now need an almost complete recontextualization. Perhaps E. Tunkelo’s Vepsän kielen äännehistoria (1946) could use some related updates too. What Pahomov’s work shows is that the Ludian and Veps varieties should be analyzed on one hand together, not separately; and that we should probably attempt a reconstruction of their last common ancestor as well. I will follow Pahomov in using the name “Old Ludian” for this.
A reconstruction of Old Ludian would probably be particularly interesting from a lexical point of view: e.g. how many Germanic loanwords have definitely made it this far by direct inheritance and cannot be treated as mediated by Karelian later on? How many exclusive shared Slavic loans are found? What unique derivatives or semantic shifts are there around? Questions of this sort will be somewhat hard to answer in detail as long as there is no dialect dictionary of Veps though. For Ludian there exists a sizable dialect dictionary Lyydiläismurteiden sanakirja (Juho Kujola, 1944), but on a closer look it is actually heavy only on the northern and central varieties that earlier research calls “Ludian proper” (varsinaislyydi), versus fairly light in the coverage of the southern, more transitional-to-Veps varieties that feature strongly in Pahomov’s argumentation. (He lists several example lexical isoglosses around pp. 163–166 though, but without clearly distinguishing innovations from retentions.) We can hope there to eventually be a dictionary of at least Kuujärv Ludian, the southernmost and today the most viable variety. Still at mere triple digits of speakers though, but aluckily including well-educated language activists like Pahomov.
But I think the new perspective on Ludian would likely force a few phonological reanalyses as well, especially if also keeping an eye back all the way to Proto-Finnic. I cover in the rest of this post two candidates.
1. Final vowels
Apocope is one basic feature that demonstrates well the heterogenicity of Ludian. Generally, all original word-final vowels are lost in northern Ludian, as also in all of Veps; in central and southern Ludian, non-open vowels are partly preserved, while final *a *ä are reduced and surface either as /ə/ (the probable intermediate stage before loss) or as /u/ ~ /ü/ (as in Olonetsian; this has been explained as a fortition from *ə due to the influence of an Old Karelian superstrate).
Original preconsonantal vowels are however uniformly preserved, including vowels followed by Proto-Finnic final *k. Yet, *-k has itself been lost everywhere in eastern Finnic. The fact that a contrast regardless remains in northern Ludian would at first look seem to demand reconstructing preserved *-k for Old Ludian. A few near-minimal pairs from the Sununsuu dialect:
- *-ak > -a: PF *polkëdak > poɫgeda ‘to tread’
- *-a > ∅: PF *valkëda > vaɫged ‘light, lit’
- *-äk > -ä: PF *imedäk > imedä ‘to suck’
- *-ä > ∅: PF *pimedä > pimed ‘dark’
- *-Ek > -e: PF *lähtek > lähte ‘spring’
- *-i > ∅: PF *tähti > ťiähť ‘star’
But this is not the only option, and does not actually seem like the most parsimonious approach.
I would suggest that rather than forms like *polkedak, *lähtek, a good starting point would be a contrast between lax and tense vowels: *polkedà, *lähtè versus *valgedă, *täähtĭ. This finds some degree of evidence already from the central and southern varieties of Ludian, where some records do show reduced final vowels in bisyllabic stems. E.g. Kortaš akkᴇ̆ ‘married woman’, Nuomoiľ & Teru Priäžä ehtᴇ ‘evening’, Viidan & Kuujärv buťkɪ ‘umbellifer plant’, with devoiced ᴇ ɪ (= IPA [e̥ i̥]), and explicitly marked as short in the first case; but e.g. Kš TP lähte, V Kj lähtə with no devoicing.
An IMO still more convincing parallel is provided by Natalia Kuznetsova’s recent research on Ingrian, most prominently in “Evolution of the Non-Initial Vocalic Length Contrast across the Finnic Varieties of Ingria and Adjacent Areas” (2016, Linguistica Uralica 52(1)), where she demonstrates that it is exactly through this path that some Ingrian dialects end up with the apocope of some but not all final vowels, and where reduced vowels can be observed to be devoiced next to voiceless consonants.
Devoiced reduced vowels have been reported from a third Finnic area showing apocope, too: Southwestern Finnish, cf. Ojansuu (1901: 24–25, 195–197). He however suggests a different mechanism for explaining the retention of final vowels in words that had PF *-k (and also *-h) — presence of a closed syllable “in some cases”. I presume this means mainly sandhi, so e.g. *läktek_meccässä > *lähðem_meθθäsä̆ > lähre mettäs [-mm-] ‘a spring in the woods’. This seems generally possible too, and may be to some degree complementary with the reconstruction of reduced vowels, but this would require an additional general survey of sandhi effects in the involved Finnic varieties.
I would propose reduced vowels can be furthermore connected with the fact that even standard Finnish shows allophonic V2 length difference between words of the shape CVCV and CVXCV. This allophonic pattern surely been also present already in Proto-Finnic, explaining why apocope with an identical counterintuitive restriction to CVXCV wordforms arises in all of Estonian, Southwestern Finnish, Ludian–Veps (ancient in these three), at least one variety of Tver Karelian (more recently), and in some dialects of Ingrian (no earlier than 19th century). Of course some of these may be areally connected, to each other or to common contact languages, but if given a “preadaptation” within the common prosodic system inherited from Proto-Finnic, they do not all need to be.
2. Consonant-stem infinitives
Most of Ludian differs from Veps in not undergoing much syncope at all. Pahomov suggests a few exceptions however, including the infinitives of some d-stem verbs: anda- : ant(t)a ‘to give’, kanda- : kant(t)a ‘to carry’, ruada- : ruat(t)a ‘to toil’; though also andada, kandada occur in central Ludian.
This is syncope alright; but it does not seem to be a specifically Ludian phenomenon. As is know, though perhaps not widely, forms such as *antadak must have had already in Proto-Finnic syncopated byforms such as *attak.  This is shown e.g. by Old Finnish infinitives such as lentä- : letä ‘to fly’ < PF *lettäk < *lent-täk < *lentä-täk, lähte- : lätä ‘to leave’ < PF *lättäk < *läkt-täk, tietä- : tietä ‘to know’ < PF *tiettäk < *tietä-täk, not derivable within the specific phonological development of Finnish. In particular the simplification of *ntt to *tt (as seen in the first) is clearly no longer productive. The restriction of consonant-stem infinitives to A- and e-stem verbs but not i-, O-, U-stem verbs moreover clearly suggests that it has arisen earlier than the pan-Finnic development of unstressed *i, *o, *U from *əj, *Aw, *əw. While no **ata, **kata are attested in Finnish (only the regular vowel-stem antaa, kantaa) or anywhere else that I know of, this is likely to be simply due to these forms falling away to oblivion: after all modern Finnish today only knows one formation of this type, tuta ‘to know’, and then only as a fossilized relic in expressions, while the productive infinitive is exclusively tuntea.
Directly inherited infinitives of this type are actually widely found in Ludian. This is not a large surprize, since across the eastern Finnic area they have been reported sporadically from Olonetsian and productively from Veps already since Setälä. Besides ruat(t)a, perusing LMS turns up among bisyllabic d-stem verb roots also at least the following:
- kieldä- : kielt(t)ä ‘to deny’ (< PF *keelt-täk)
- kiändä- : kiät(t)ä ‘to twist, turn (tr.)’ (< PF *käättäk < *käänt-täk)
- kuada- : kuat(t)a ‘to pour’ (< PF *kaat-tak)
- lendä- : let(t)ä ‘to fly’ (< PF *lettäk < *lent-täk, cf. above)
- löudä- : löut(t)ä ‘to find’ (< PF *leüt-täk)
- (? nouda-) : nouta ‘to follow’ (< PF *nout-tak)
- püuda- : püut(t)a ~ püudada ‘to hunt’ (< PF *püüt-täk)
- siädä- : siät(t)ä ‘to do’ (< PF *säät-täk)
- sorda- : sort(t)a ‘to fell’ (< PF *sort-tak)
- souda- : sout(t)a ‘to row’ (< PF *sout-tak)
- tiedä- : tiet(t)ä ‘to know’ (< PF *teet-täk)
- tunde- : tut(t)a ‘to know, feel’ (< PF *tunt-tak)
- vierdä- : viert(t)ä ‘to burn extra wood at a slash-and-burn field’ (< PF *veert-täk)
- viändä- : viät(t)ä- ‘to dance; to bend’ (< PF *väättäk < *väänt-täk)
- uurda- : uurtta- ‘to carve’ (< PF *uurt-tak)
In this light, also ant(t)a and kant(t)a are unlikely to represent irregular late syncope from andada, kandada: they should be instead considered analogical reshapings of inherited *atà, *katà! with simple reintroduction of -n- from the vowel stem. The same analogical reintroduction of -n- is found also in ‘to plow’ (kündä- : künt(t)ä ~ kündädä), ‘to push, send’ (tüöndä- : tüönt(t)ä); and without gemination, a similar analogical reintroduction of -h- is found in ‘to leave’ (lähte- : lähtä ~ lähtedä).
In tA-stem verbs though (i.e. in those original *tA-stem verbs where the preceding consonant was voiceless, preventing voicing), only vowel-stem infinitives seem to occur: (-ht-) ahtada ‘to set up’, kiehtädä ‘to bother’, puahtada ‘to roast’, puohtada ‘to clean grain’; (-st-) kastada ‘to dip’, kestädä ‘to stand’, nostada ‘to lift’, püštädä ‘to stick’; (-tt-) ďiättädä ‘to leave’, keittädä ‘to cook’, ottada ‘to take’, suattada ‘to accompany, transport’; (-ɫtt-) poɫttada ‘to burn’.  This probably means that the origin of OFi. lätä, Lud. lähtä is to be dated to an even earlier stratum than the Proto-Finnic reduction of *A in the context *t_t. As is quite probable: Proto-Uralic *läkt(ə)- is also e.g. the only consonant-stem verb in Mari that ends in a cluster of two voiceless consonants, and the only consonant-stem verb known in Udmurt at all.
Lastly a second regular vowel-stem infinitive group consists of iďädä ‘to germinate’, pidädä ‘to keep’, vedädä ‘to pull, draw’: reduction and loss of unstressed *-A- following a light stressed syllable is not expected/precedented in any morphological context at all.
Neither of these examples really ends up changing much about the reconstruction of Proto-Finnic itself. The first is on the phonological level indeed simply a “patch” for the development path of apocope, though it is also one piece of evidence for the reconstruction of allophonic vowel length. The second seems to provide the first attestations pointing even indirectly to the infinitive forms *attak, *kattak, though examples like letä, tuta also already allow hypothesizing such forms anyway. But the take-home seems to be that while the segmental phonology and morphology of PF are well-known by themselves, two areas suitable for further work would be prosody and morphophonology. Both of these incidentally also become much less charted territories when looking further back towards Proto-Uralic.
 At minimum; it is conceivable that the vowel-stem infinitives are altogether later analogies.
 Old Finnish too seems to show no examples of consonant stems for verbs in -htA- or -sta-. The few examples for verbs in -ttA- could be themselves analogical, sometimes even misinterpretations.