Theory
The best time for forced relocation of populations by social characteristic (e.g., national homogenization) is in the immediate aftermath of war. The opportunity cost is at its lowest because the destruction and chaos has already disrupted the economy. National borders are being redrawn, and new nations are being born. Most importantly, the general opinion of multiculturalism is at an all-time low at the end of a war, while nationalistic fervor is high.
The Caucasus -- OTL
That's certainly how things looked OTL after WWI. In OTL, though, while Europe got two relatively humane forced relocations after WWI & WWII, the Caucasus got a disaster -- poorly drawn borders betrayed mixed motives, and the hallmark event was the Armenian Genocide.
The Caucasus -- XXth C
It doesn't have to be that way. In XXth C, we have a very different war.
One where the Caucasian nations play a brave and proud role -- defining themselves in war and in diplomacy, and enbarrassing Russia.
One where Turkey doesn't totally embarass themselves -- and therefore never launches the cultural juggernauts that were the Young Turk and Ataturk's Republic eras.
One that ends without Mandates and without Soviets.
Proposal
The nations that stand to gain, in my mind, are Georgia, Armenia, and Azerbaijan.
- Georgia is in a position to win the Circassians and Chechens into their national orbit. In the long run, this could be reckless (Orthodox-Muslim relations are tricky), but unless Georgia and Armenia unify, that's the direction that they would expand in.
- Armenia I posted what they want above. Armenians were spread out over a pretty large area at the end of the Ottoman Empire. A really generous settlement would also give them the Nineveh plains, where the Assyrian Christians live. Since as I recall, Turkey didn't collapse at the end of the war, though, I don't see Armenia getting everything that they want. I do see Armenia ending up with the lands that are co-populated by the Azeris.
- Why? Because the Azeris are going to do very well for themselves regardless. Russia and Persia are both a mess, and the new Turko-Arabic Federation doesn't have much of a legitimate claim to their lands. On this map of (modern OTL) Persia, I would award Azerbaijan all of the lands marked "Azeri". Now in time, Turkey might be interested in unifying with the Azeris, who are arguably their closest ethno-linguistic partners, but that's a story for another day.
Caveat
It's been a while since I read through XXth C, so I might be misremembering a lot of details...