I believe I am descended (on one side) from the Khazars. The eyes never lie.
The galut of the ten tribes was not absolute, there were remainders.
It is plausible that the galut of the capital city included many visitors or migrants from the outer areas and that samples of the 10 tribes went to Bavel, and came back with Ezra, or stayed whioh became the eastern diaspora that became the Mizrachim when Spain emptied out. One can also try to make sense of the end of Ovadyah, which speaks of two galuyot, beni yisral , and yerushaliyim. Survival was the aim of the remnants and tribe in galut was immaterial. The close gene pool of the Ashkenazim indicate that it may have been a small remnant indeed that became the Roman driven nucleus of Ashkenaz. In the future, the tribes must be accounted for to have shmitta and yovel, and a reapportionment of the whole land.