Horse crossbows are deadly against HA in the right situations. Back when HA were powerful instead of the skillbased situational class it is now, as infantry support cav I had to carry a light crossbow just so I can point it at the HA to ward them away, since showing a crossbow caused many to take evasive action. If they wanted to call my bluff, the light crossbow was accurate and damaging anough to hit them or the horse with 0 wpf. It was especially deadly in the gun to head shots when one is making a close range high speed pass, often resulting in an instant kill or dismount, which to a horseman is pretty much equivalent. Not to mention, nimble lancer giving you the headache? Pull out the crossbow and blast his light horse, that will teach him.
Pure cav who do not wish to carry infantry only weapons can outfit themselves with lance, shield and light crossbow to deal with all the dangers of mounted combat in 4 slots.
HA versus HC fights are quite entertaining. The Ha does not want to get into close range drive by shots, where the crossbow will have the advantage. The HA also wants to shoot the HC while he is reloading. So HC will chase as HA wants to keep the fight at high speeds, then HA will chase the HC while HC is reloading. Close range shots is usually a tossup, but the HC can cause massive damage if he hits. Do not trot about while fighting the HC since the HC will win, and HA's higher rate of fire allows more adjustments while shooting at top speed.