There are a lot of theories about how that happened. One is that the native languages in the prehistoric period (in that part of the world) were actually ancestral to the Turkish language(s) that were spoken by later Eastern invaders.
What to look for in the future as we learn more and more about this phenomenon ~ could be the spread of the X-Factor DNA sequence. This is a DNA sequence that acts more or less as a bar-code for the Sa'ami. People who descend from the Sa'ami ~ a people who moved out of the Western European Ice Age "refugia" ~ of about 14,000 years ago, or MORE, have this sequence.
So far the Fulbe, Berber, Yakuts, Sa'ami, (lots of Finns and Norse), Iroquois, Chippewa and Cherokee have a large percentage of their number who have this sequence.
It is known to have originated among the Sa'ami and spread to the other groups.
Thank you for the info on Asian migrations...