That tactics would be different. There are bonuses to the crossbow, namely that it is easy to use and can cut through armor fairly efficiently. But to properly use a crossbow on horseback, you'd have to slow down... I think. Perhaps the Mongolians could have pulled it off at full gallop... I have heard their best riders could hang off the side of their horses so that the horse's body was between them and their enemies, where they would return fire from underneath their horse's chin. Those guys knew what they were doing. 

Still, the imagine I can't help but have of mounted crossbowman is as a mounted firing line. It's a legitimate tactic, just not the one people assume when they think of horse archers.