How did Byzantium lose it's grip on Anatolia?

It seemed like for the longest time before 1075 A.D the Byzantines had controlled this region militarily through use of the "Theme" system, while also demographically controlling it, as Pontic Greeks had firmly settled the area and dominated it ever since the invasions of Alexander the Great and early Greek colonization of the area by Merchants hundreds of years before that.

So my main question is; Did the demographic replacement of Greeks in Anatolia happen over a long period of time through a slow process, or was Greek migration out of Anatolia triggered by the Seljuk or Mongol invasions?

It's evident based on historical records that these areas of Greek settlement, such as the Crimean peninsula and other areas on the Western part of the Pontic Steppe did experience a shrink in the native Greek population over time. Pontic Greeks in particular were shrunk all the way down to Trebizond, yet they once dominated the whole area of Pontus.

I guess this could be compared to traditional Eastern Slavic people fleeing to what today is Moscow after the sack of Kievan Rus' by the Mongols. However since Anatolia was heavily under Greek and Hellenic influences for over a thousand years it seems unlikely that a single invasion(Mongol or Seljuk) would trigger a massive exodus on that enormous scale.

Any thoughts on this? I'd like to hear what you guys think.

Assimilation for the most part in my opinion. This happens a lot in history actually. After the Turks broke Byzantine power at Manzikert the Turks overran much of Anatolia. Keep in mind, however, the Byzantines didn't just give up on the region after Manzikert. The Byzantines managed to recover parts of Anatolia and lose them again over the next 400 years the problem was that many other factors forced Byzantine attention away from Anatolia like affairs in Europe. When the 4th Crusade torched Constantinople they pretty much shattered the empire and it took the Byzantines many years to retake Constantinople and recover parts of their empire in the meantime the Turks had centuries to assimilate the Greek populations which just happened overtime. That also doesn't mean the Greeks fizzled out either. There are records that show that Greek merchants continued to thrive in Anatolia after the Turkish conquest but with the Turks being the dominant culture in Anatolia for so many years the Greeks either assimilated or migrated away. This is very similar to how the Romans operated. By the time of the Roman decline many of the native peoples they conquered like Gauls, Britons, Iberians, Illyrians, etc either died out or adopted many Roman traditions to the point that they weren't like their ancestors at all

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