I’m not sure. If the divergence between Middle Eastern and European Jews occurred about 2,500 years ago, then it would seem arguable that the European Jews are descended from the 10 tribes that were carried off by the Assyrians and allegedly scattered into the Caucasus and parts of Central Asia. In any event, it is all interesting.
I would tend to suspect that to be the case. The Khazar story, on the other hand, says that the Khazars - a Turkish group - adopted Judaism wholesale in the 8th or 9th century, so as to establish their independence from both the Christian Byzantine Empire and the Islamic Caliphate. This much is the case, if I'm recalling my history correctly. However, what is usually added to this is the largely unsubstantiated belief that the Khazars themselves (instead of examples of Jews deported to Assyria) became the European Jews. Usually, the context I've seen this in is as part of an effort to deny to the Ashkenazi (and Sephardic) Jews any lineage from Israel, the idea being that today's Jews are "usurpers" who are really just Turkish Khazars.
Nope, the 10 tribes are still lost. All of today's Jews are descendants of the Southern Kingdom (Judea).
It was almost exactly 2500 years ago that the Kingdom of Judea was conquered by the Babylonians and all of its inhabitants exiled to Babylon. About 70 years later, the Persians conquered Babylon and permitted the Jews to return to Judea, but most did not.
Today's Ashkenazic Jews are descended from those who returned to Judea; when Judea rebelled against Rome in 70 C.E. and again in 135 C.E., the Judeans were taken to Rome as slaves. Their descendants were permitted to move from Italy to Germany by Charlemagne, and from there they spread to Poland, Russia and eventually to the United States.
Meanwhile, the Jews who did not return to Judea stayed in Babylon (today's Iraq) until it was conquered by the Moslems; they then moved throughout the Moslem world (the Middle East, North Africa and Spain). Their descendants became the Sephardim (Jews exiled from Spain in 1492) and the Mizrachim (Jews who remained in Middle Eastern countries until the Arabs expelled them in 1948).