First. The northern kingdom, "House of Israel", along with many from the southern kingdom, "House of Judah" were taken by Assyria in 745-721 BC, well over eight hundred thousand Israelite men 2 Samuel 24:1 - 9 and their families from ten different tribes (2 Kings 17, 18, and 19). In 2 Kings 19:32-34 we learn that only Jerusalem was spared. Vast numbers of these Israelites, soon after the power of the Assyrians was broken, began several migrations, with the main two groups moving west under the Black Sea, and north through the Dariel pass of the Caucasus mountains into the steppes of south Russia.
A large group also moved east. These were called Sakka (Saka) and Iskuza by the Medes and Persians. The Japanese name Sakai is but a step away from Saka. There are many strange customs of unknown origin in Japan that can only be explained by recognizing that some from the east Jordan tribes of Manasseh, Reuben and Gad, Manasseh being pre-eminant, made their way to that far eastern Island, while their cousins headed west to populate northwest Europe and the far western island of Britain and later on into the Americas.
Some of these migrations were undertaken all the way into the 17th century when, in the final migration, some from the tribe of Manasseh sailed the north Atlantic to Plymouth to fulfill Isaiah 49:20, which was prophesied to the hegemony of the House of Israel, some from the tribe of Ephraim (England).
You see, the crux of the whole Lost Tribe subject is that the names we want to look for to trace the Israelites are not the names that historical accounts and archaeological finds give those same folks.
To more confuse the issue, large groups of Israelites called themselves by different names. Some of them called themselves the House of Isaac, which is pronounced e-sahk with the emphasis on the last syllable. How natural for the Persians to call them the Sakka (Sacae in Greek), while the Assyrians called others, the House of Omri, after the sixth king of Israel. This name sounded like Khumri, and was variously pronounced Ghumri, Gimri, Gimira, Gammer, all of which turned into the Greek Kimmeroii, our English word Cimmerians.
The Israelites weren't lost, their name got lost. That fact coupled with the erroneous search for the Jews' fulfillment of the Old Testament prophesies has held the Lost Tribes teaching in virtual obscurity these millenia since 500 BC.
and Second. from the southern kingdom, "House of Judah" one hundred and thirty five years later, in 586 B.C, Nebuchadnezzar of Babylon carried more than one half million Israelite men 2 Samuel 24:1 - 9 and their families from the other two tribes, of Benjamin and Judah (the Jews), away to Babylon. 2 Kings 25:1 - 21.In 457 BC King Artaxerxes decreed that any Jew in his realm, including the priests and Levites, may return to Jerusalem Ezra 7:13 after the Temple is rebuilt. Ezra and Nehemiah went back to Jerusalem, but they took with them only a REMNANT, and only from the House of Judah.
The rest of "Judah", the majority of them, went north over the Caucasus Mountains with the other tribes from the "House of Israel" and to this very day most of true Judah and Israel do not know who they are. Though they basically comprise the Christian nations of the world, in their ignorance they call themselves gentiles.