about Christians of the Orient fighting with Muslims against whatever.

Ignore the past history which might not be correct all the time according to the books were written about it.

In modern history which we know very well and parents and grandparents lived it. The Christians were fleeing the military service within The Ottoman army at least in Lebanon and Syria. They were immigrating to various places of the world for that reason. Some arrangement was reached later to except the Christians from joining the "Islamic Empire" army. I don't know what was the cost of that exception