It is quite weird that european cavalry in raiding, scouting and foraging and other situations did not adopt bow.
On the other hand it did adopt crossbow in several occasion, especially auxiliary.
But, basically, as long heavy cavalry is the marker of social distinction, using a weapon as poorly considered socially as the bow isn't really going to be a thing.
Horse archer best asset is that they can kill and wound horses quite easily.
Mongol horse archery (which quite perfected it) was more about saturation than searching individual targets.
For an horse archery to be that efficient on battle, you need not only huge discipline (would it be to pull tactic such as feigned flight without the whole thing going to an actual flight) but huge numbers to saturate the battle.
Even there, Mongols suffered important losses at Legniz, even in face of tactical and numerical superiority, so the "quite easily" may have to be really nuanced.
Men are harder since they tend to armour themselves.
Horses tented to be armoured themselves, at least since the XIth century and the rise of siege/raid warfare (first leather parts, with more important armours and protection being adopted in the XIIth century, after the Crusades' experience of light cavalry harassing).
That's assuming they are poorly trained horse archers.
Training a massive horse archery is quite hard actually, critically without such military tradition.
First you'd have to make it the main military force (would it be only to allow saturation tactics) against the aformentionen cultural/social bias (to say nothing of the absence of a real motivation in a warfare essentially based on sieges)
Then training the main part of your army to fight on horse : even in medieval armies, it never represented more than half and in pretty exceptional conditions. And that while training to use a bow skillfully in the same time. (There's a reason why horse
cranequiniers were a thing, while mounted bowmen were essentially mounted infantry)
Of course, it requires enough discipline to allow basic tactics (as "feign flight" not actually turning in a full-fledged retreat) and to keep the cohesion of your army.
So, I really think it would require much focus (without actual motivation doing so) to properly train a useful western horse archery that would replace heavy cavalry and that without social, tactical, traditional support (at the contrary : pulling it out of nowhere have really few chances to ever works)