The horses bred by the most effective mounted archers were used to
grazing as opposed to horses that needed grains to feed. Thus the
armies of the Huns, Magyars, Mongols, etc. did not need carts to carry
food for their horses, making them more mobile than other armies.
Moreover, European archers used crossbows, which were very slow to
reload (2-3 shots a minute as opposed to 15-20 shots) and impossible
to use while riding a horse.