I'm talking about the rise of the Seljuk and later Ottoman empires. Were non-Turkic people displaced or were they gradually assimilated into Turkic culture?
Were Turkic people even interlopers or had they lived in Anatolia for some time as well? Am I asking the right questions?
Im just going to copy and paste my previous answer.
This question is sort of answered here:
A lot of Anatolia became Greek even before Alexander (550-350 B.C.) most of the coastal cities in Anatolia were Greek even long before that and Greek (culture, language) spread through out Anatolia after the end of the Lydian empire. It stayed Greek through Rome/Byzantium. The Swedes, Rus, other slavs and other Orthodox Christians that settled over time eventually became Greek (culture, language etc). Later those Greeks became Turks and the vast majority of Turkey's population were native people in Anatolia who converted to Islam, rather than the actual Turkish tribes and other Islamic peoples who settled there.
The reason to convert to Islam is to not have to pay the Jizya (a tax only Non-Muslims called Dhimmi had to pay), they also converted to not be discriminated against or they converted to identify with the victor. Some may have also converted because they liked the culture and faith better than Christianity or Judaism.
Also to further answer your question a lot of the Christians (Greek mostly, and others like Armenians who were Orthodox etc.) that were in Anatolia and did not convert to Islam, went to Greece during the 1923 population exchange between Greece and Turkey.
You can get a brief account of it in chapter 14 of "The Middle East" by William Ochsenwald & Sydney Nettleton Fisher.