The Crusaders generally saw the Christians of the Middle East as having local loyalties that trumped whatever religious affinity they might have for the invading Catholics. It is important to note that there was quite a bit of enmity between the different Christian sects, especially in light of the recent schism of the Orthodox and Catholic churches and subsequent events such as the Massacre of the Latins. At the same time, political pragmatism often trumped religious loyalty in the conflicts that followed the First Crusade, and there were instances when Catholic crusader states allied with Sunni emirs in order to fight other Catholics (who might have their own Sunni allies). In such a climate, it was not so hard to believe that a local Miaphysite follower would stick with their Muslim neighbors against the Catholic foreigners.

Ironically enough, the Muslims had quite the opposite worry- that religious ties would be more important than secular relations, and that their Christian subjects might plot with the Catholics in order to facilitate the conquest. This led to situations where the Muslims would persecute Eastern Christians when they were under threat from the crusaders, to the point of evicting them from cities under siege for fear that they would collaborate with the attackers, despite the fact that the Catholics had no such intentions and the Christians in question just wanted to be left alone.

Source: The Crusades through Arab Eyes, by Amin Maalouf