I think not. Most people don’t flee their homes or die fighting when their country is conquered. Spain was majority Christian and had a large Jewish population all during the period of Muslim rule, likewise in all the countries of the Balkans after the Turkish conquest — even the Muslim Albanians and Bosniaks are descendents of Christians who stayed. Egypt was majority Coptic Christian for centuries after the Muslim conquest.
Facts.
I will do some digging this week on this subject and let you know what I find.
That said, you might be right. Or you might be wrong.
Fleeing wasn’t nearly as practical 1000 years ago as it is today. The Middle East is rapidly losing its minority religious groups for the simple reason that they have places to go.
1000 years ago, not so much. A peasant who fled lost everything and went into a land where he didn’t speak the language and had no way to earn a living comparable to what he had before fleeing.
However, quite a few did flee into Byzantine lands and were settled by the emperors along the borders as peasant/soldiers, the rough equivalent of the later Cossacks. The Habsburgs did much the same along their Military Frontiers.