I would suppose they had less reason to stay and fight, but not much less. Soldiers at times didn't understand why they were a gazillion miles from home, attacking some people for no tangible profit of their own, particularly non-professional soldiers I would imagine.

Being under your command however, I believe I would be deserting before long. Noone likes being seen as second-rate, and as the mercenaries were better trained fighters than most of the armies they took part in I would expect that they were considered valuable by their employers. I'm no historian though.