A small definition of a concept.
Across the dialectology of various languages we very often find almost the same segment inventory despite various innovations. I call this phenomenon “phonological renormalization”. It seems somewhat mysterious at first: it is hard to see any way how a language’s status as a part of a large dialect continuum could outright prevent innovative phonological features from arising. However, it does seem to me that there could be an easy way out — by assuming a slight diachronic detour: suppose that new innovations do arise, they simply then afterwards change back into another segment already known in the language’s other dialects. A sufficiently homogeneous “sociophonetic environment” could probably motivate novel phonological segments to often merge with pre-existing close matches. Specifically, learners / innovating speakers faced with the prospects of (1) adopting an innovative form and (2) adopting an innovative segment might prefer the former, but still avoid or fail to manage the latter.
This kind of hammer-down-the-nail development can be sometimes directly attested. Both stages are historically recorded e.g. for the fate of *ð  in some western dialects of Finnish: first flapping, to create a new phoneme /ɾ/, then merging with the usual trilled rhotic /r/. Or, in Eastern dialects of Finnish: early loss of medial *-n- in the allegro forms of the 1PS and 2PS pronouns minä, sinä evidently first created a rare transient diphthong /iä/, recorded only in a narrow southwestern corridor from Mikkeli to Hamina; but elsewhere renormalized to /ie/. Even in Mikkeli it has been fortified by the late Savonian–Karelian diphthongization of *ää to /iä/. I also wonder if the Western Finnish allegro forms mä, sä actually continue older miä, siä with different renormalization (but before the lowering /ie/ > /iä/).
My aim is not to list extensive amounts of evidence here. Renormalization is pretty easy to notice once you start looking, maybe especially among consonants and/or small phonological inventories? But just off the top of my head, a few other conspicuous examples that come to mind include the following:
- Glide epenthesis in Finnic and Samic. Both language groups generally turn *w into labiodental /v/ = usually an approximant [ʋ]. Whenever any kind of a labial glide develops later on, e.g. before word-initial /oː/, or in hiatus following any labial vowels, or from lenited *b, this likewise tends to produce /v/, not [w]. Occasionally a [w] can be attested, e.g. in Standard Finnish in cases like *kauɣan > kauan [kau.an ~ kauwan] ‘for long’; for which maybe most dialects however show instead kauvan.
- Preaspiration *TT > *ʰTT in Samic. This affects first the native Uralic geminates already in Proto-Samic; slightly after that, across western Sami, also secondary geminates introduced by consonant gradation.
- Lenition-plus-fronting *ɣ > /v ~ j/ in Mordvinic (/v/ in back vowel environments, /j/ in front). This affects first *ɣ from PU medial *k across the whole group, slightly after that also *ɣ from the lenition of *ŋ, across all of Moksha and most of Erzya.
At least the second has been even proposed to actually represent a single innovation, which would postdate *Cˑ > *Cː in western Sami but predate it in eastern. An equivalent scenario, with old *w and *ɣ preserved until the rise of secondary cases, could be sketched from the other two too. But perhaps the phenomenon of renormalization should be a sufficient explanation.
In more general, I suspect this phenomenon could perhaps end up accounting for much of what is normally called “cyclicity” of phonological rules. In a language that e.g. aspirates its initial stops (say, English) it’s not that a newly born /t/ from a source that does not already have aspiration (say, from /θ/) would have to be instantly aspirated, since we do sometimes find allophonic contrasts phonemicizing in this fashion. But if this did create a three-fold contrast /tʰ/ : /t/ : /d/, I predict that this should prove unstable and be quickly followed by an additional merger [t] > [tʰ]. After this the innovation could also no longer spred to other speakers as a “phonetic” change [θ] > [t], only in a purely phonological form /θ/ > /t/ [tʰ : t], potentially quickly leaving any small original areas of /tʰ/ : /t/ in obscurity.
And I wonder further… other kinds of reiterated sound changes could find similar explanations for them too. Juliette Blevins, for example, has a recent paper observing that Austronesian *q has disproportionately few reflexes that are actually /q/ She proposes that this should be taken as evidence that the phonological stability of uvulars is conditional on a language’s vowel system, which might not be all wrong. (Cf. some further comments from me @ Tumblr.) But could it be additionally the case that an area having many languages with /ʔ/ creates a pressure for other languages to “normalize” (hardly “re-“) even an inherited, native *q into /ʔ/…? Or, indeed, even a *k? setting thus the stage for the famously common-in-Oceanic chainshift *t *k > /k ʔ/.
Of course though, this goes both ways and it’s also possible that many cases could be accounted for simply by internal phonological instability.  Already in the above examples: e.g. a contrast [w] : [ʋ] would be itself pretty rare, ditto the western Finnish system with all three of /ɾ r rː/. Still, at least the specific direction in which these systems seem to collapse does look to be largely determined areally, and e.g. no cases of the chainshift *r *rː > /ɾ r/, known from places like Ibero-Romance and Albanian, has been described from any western Fi. dialect.
 Reminder for the map reader: Kettunen’s ð is usual Finno-Ugric transcription for the alveolar tap = IPA /ɾ/, while δ is the voiced dental spirant = IPA /ð/.
 Reminder thanks to my wife Sara Carrier-Bordeleau.