While it was common for most mounted crossbowmen to dismount prior to shooting their weapons, there certainly also were some that stayed mounted.
Granted, they weren't western versions of horse-archers, but would rather be well-armored, and participate in the mêlée.
Hans Thalhofer, a German fencing and general weapons expert from the 15'th century wrote several books on weapons use and warfare, one of which apparently should have a complete chapter on mounted crossbow use. Unfortunately I haven't been able to find that book, though there is a single picture here, in another book: http://www.kb.dk/da/nb/materialer/ha...ott-2_290.html
Phillip Von Seldeneck (also 15'th century) writes in his Kriegsbuch (War Book) that the mounted crossbowmen should be placed on the left flank, and attack the enemy lancers right after they had lowered their lances, because the enemy would usually carry their shields to their left, and they can hardly defend themselves with their arms.
He also specified that they should fire only once, (Which would also appear to indicate that they were capable of shooting several times). and then immediately engage with their swords, either trying to chop the tips off the lances, or to wrest them away from the lancers. If possibly they should then penetrate deeper, and engage in hand-to-hand-combat.
By this use, their main purpose was to break up the enemy lancers formation, and due to the immense danger of such actions, Phillip dubbed them 'doomed marksmen'. They should work in tandem with friendly lancers who would engage the enemy as soon as the crossbowmen had succeeded in disrupting the enemy formations.
The formation can be seen here; Mounted Crossbowmen in front, with Lancers in the back: http://diglit.ub.uni-heidelberg.de/diglit/cpg126/0105
As such, they filled much the same role that cavalry with hand-cannons later would do.
With the invention of full-metal crossbows, and push-lever and ratchet drawing mechanisms, made it possible to have powerful and reload-able crossbows that you could use while mounted. However, it seems likely that they would slow down/stop when reloading.
Crossbows could also be used from horseback when hunting, like Emperor Maximilian does here:
So, you're right in that they're not depicted properly in the game, but mounted crossbowmen did exist.