Although culturally Turks prefer the middle asia hun and monolian origin narrative, they are genetically mostly descendants of the anatolians who happened to live and breed there already for many millenia. Those people didn't suddenly disappear. This contrast between culture and genetics is confusing to many turks (me included)
That's what I think happened to other countries in the region as well. I'm from Iraq and speak Arabic but I don't think my ancestors are from the "Arabian peninsula", we just adopted the language. It also explains why "dialects" of Arabic can be so different.
Well not Greeks (in the main), but Anatolians I guess. Certainly not Central Asian.