Recently I sat down with my copy of J. Lehtiranta’s Proto-Samic dictionary, Yhteissaamelainen sanasto (1989; SUST 200) to work out the development of the vowel systems in the Eastern Samic languages. I do not know if this has been done before; it might have, though I am not exactly worried about rederiving results.  At minimum this topic is absent from Sammallahti’s handbook The Saami Languages (1998). His historical phonology appendix covers at length only the evolution from Proto-Uralic to Proto-Samic, and from there to Northern Sami. In the main chapters, too, he only mentions a handful of innovations for Eastern Samic (that he deems diagnostic for defining its taxonomy). Yet it’s obvious that there’s been much divergence going on here: cf. e.g. *kōlē “fish” > Inari Sami kyeli, Skolt Sami kue´ll, Kildin Sami kū´ll, Ter Sami kïĺĺe. 
The following fairly general features stand out:
- The umlaut tendencies that already must have begun in the Proto-Samic era have continued wildly. Most vowels have distinct reflexes before each of the three common PS stem vowels: *-ë, *-ē, *-ō. (Since Lehtiranta only lists citation forms of words, I don’t have much idea what the effect of PS *-ā, *-i, and *-u, which are rare outside of inflected forms, has been.) As usual, this must’ve been allophonic at first, but was later widely phonemicized by loss of unstressed vowels.
- The mid vowels *ē, *ea, *oa, *ō, *o, *ë have the most varied reflexes. The close *i, *u are mostly unaffected (only Skolt has any umlauts going on with these), and *ā has not been majorly affected either.
- Although not all languages distinguish all different umlaut “grades” of various PS vowels, I suspect umlauts for the most part regardless occurred in Proto-East Samic already, and that various languages have simply secondarily lost certain distinctions — since they seem to have done so in different ways. E.g. in Inari, *ë-ë and *ë-ō both > /o/, versus *ë-ē > /a/; but in Ter, instead *ë-ë and *ë-ē both > /ɐ/, versus *ë-ō > /o/. Skolt and Kildin distinguish all three types.
- In addition to the umlaut splits, there also seems to be a vowel lenght split. There is of course no sign of this in Ter, where vowel lenght contrasts have been lost altogether; but it’s found relatively robustly in the other three languages. This seems regardless a little bit more like an areal phenomenon: lengthening in Inari almost always implies lengthening in Skolt, but Kildin corresponds poorer to these, and there are also cases where lengthening is found only in Skolt. As for conditioning, long vowels seem to be the rule of thumb before singleton medials, short vowels more general before two-stop clusters. This includes geminates, so the change must have been earlier than the strengthening of the strong grade of single stop consonants to geminates in Skolt. I’ve not worked out the conditions for other consonant clusters yet.
- Skolt Sami seems to be altogether the Sami variety with the most complicated vocalism (though Southern Sami could give it a good run for the title). At its best, *ea has no less than seven different reflexes: eä, iä, iâ, iõ, ie, e, ee!
I do have a full correspondence table charted out, but further details shall come later once I’m done dubblechecking things.
All this has clearly had one important effect, though: loanwords seem to frequently “fail to keep up” with all the hair-thin split rules going on. Generally such cases seem to remain phonetically closer to the loaning language. It follows that such loans have to be dated as newer than Proto-Samic; indeed, possibly as newer than the splitting of all dialects in question. Even then, many such loanwords show a distribution across nearly all of the Samic languages. This seems to be another good demonstration of a point I think Uralic etymology needs to pay a lot more attention to: the “distributional principle” (“a word dates to the common ancestor of the languages it is found in”) cannot be trusted in the case of loanwords.
— There’s also one interesting feature that suggests some reinterpretation of the Proto-Samic vowel system. The *-ē-grade reflexes generally seem to be somewhat fronted, when distinct from the “unmarked” *-ë-grade reflexes (cf. e.g. “fish” above). On the other hand, *-ō has had a fairly general lowering effect, not so much a labializing one. This is only natural insofar as *-ō merges with *-ā in Skolt thru Ter. But it does remain a labial vowel in Inari. So what’s up with changes such as *pēŋkë > piegga “wind” vs. *pērkō > piärgu “food”; *mōrë > muora “tree” vs. *mōlōs > muálus “thawed water at shore”? And for that matter, Sammallahti notes that *ō caused also earlier lowering of PU *e, *o to PS *ea, *oa;  he posits a relatively open value [ɔː] for the vowel for this reason.
I now have formulated a different hypothesis. The etymological origin of *-ō is unclear — but most proposals have involved a coloring of PU *-a in some fashion. However! If there was indeed a change *-aw > *-o, perhaps this should be postdated to the dialectal Sami era. The following chronology seems to have potential:
- Late Proto-Samic: 2nd syllable *-a > *-ā generally changes to PS *-ē, but remains in PS stems of the shape *-āw.
- After the W/E split: Secondary *ā-umlaut in Eastern Sami.
- After further dialectification: *āw coalesces to *ō in Western + Inari Sami; but merges with *ā in Skolt + Kola Sami.
Of course, this would require looking into the consequences. One issue is that Proto-Samic had not only the traditional *-ō-stems, but also the class of *-ōj-stems. How should these be reconstructed in this system? I don’t think anything with front rounded glides (*-āẅ?!) would work, since PS had eliminated front rounded vowels from its phonology. Maybe *-āwjV?
Followups: Part 2
 If anything, I consider this a much better way to get a hang of known results than just reading about them from a reference book. Also, I did this kind of a survey on Livonian once before and that ended up with me making a couple of discoveries that have by now grown to a draft paper.
 Or, supposedly, with “light” palatalization (UPA subscript half ring), not “heavy” (UPA superscript acute). I’ve seen the similar contrast of “palatal prosody” vs “segmental palatalization” in Skolt Sami transcribed as one of secondary palatalization [tsʲ sʲ nʲ lʲ] vs. full palatality [tɕ ɕ ɲ ʎ] though — and given how the UPA is surprizingly terrible at representing primary palatals, I’m guessing this is the case for Ter Sami as well. Especially since both languages lack a “heavily palatalized ŕ” where expected, which squares well with how palatal trills are physiologically impossible.
 Actually [ɛː], [ɔː] according to him. I have a couple reasons to think these may have diphthongized already early on, though; but that’s ever so slightly off-topic for this post…