I think you got things backwards.
Asia minor (Anatolia) was always Greek The people that lived there were genetically Greeks, they spoke Greek, and had a Greek culture. The ruins are of Greek architecture, and all the inscriptions are in Greek. In fact even those who call modern turks are genetical mostly Greek.
It is the Ottoman occupiers' who do not belong to on the other side of the boarders and not the Greeks on Lesvos and other Islands who have lived there for thousands of years.
Ancestry .com who conducted the turkish DNA project, revealed that most turks are more Greek Genetical than Anatolian turks. They are simply Pontiac Greeks, Armenians, and other ethnicities of that are who survived the genocide, and were torched to convert to Islam,
Why We Put Greek and Turkish Together - DNA Consultants