I believe most hunting of this sort was conducted with lances, spears, swords. Crossbows became popular hunting weapons later in the Renaissance but I don't think the self bow was widely used by the nobility in Western EUrope from horseback, either for war or the hunt. (I'd wonder seriously if the Longbow oculd be used from horseback at all). I know some Russian knights were trained as horse archers, but I have never heard of any further west.
If there is some evidence of this I'd be very interested in to see it, it would change my understanding of western military history.
Horse archers were used in Spain during the Reconquista, primarily by Moslem forces, but apparently also by some Christian armies, particularly in Andalusia. Saracen forces based in Italy also used some horse archers. The Magyars of Hungary were famous for their use of horse archery, and the Poles used them also. You are no doubt aware that they were widely used by the Turks, who were "further west" than Russia.
But from everything I have seen, you are correct that horse archers' primary area of use was in Eastern Europe, not Western. Mounted archers were widely used in the West but they dismounted to fight.
David Craig