Mounted crossbowmen were real. According to Ralph Payne-Gallwey, they were more popular in Spain and France, whereas the English had quite a lot of mounted longbowmen. In terms of how they fought, I imagine it was probably more of the former – as with the dragoons of the blackpowder era, the mobility of having a horse can be hugely useful in terms of getting your ranged infantry to the best terrain first. But I imagine if for no other reason than self-defense, it would be useful to use bows that could be fired and cranked from the saddle.