I am a Canadian who has only had limited experience shooting guns (i.e. gun ranges in Las Vegas). However, everything about those experiences indicates to me that the notion of firing a gun while riding a horse at virtually any speed would be an exercise in futility. However, countless films & TV shows about the "western" era show characters shooting handguns and rifles on horseback to reasonable effect.
Is this just a cinematic trope that is shockingly unrealistic once given any serious thought? Or do I underestimate the skill that humans are capable of developing given enough practice?
We're people in these time periods known for actually being able to use firearms effectively from horseback, at speed?
Shooting from horseback is certainly possible, though not easy. Professional military cavalry was trained regularly in this skill up until WW2. Poland, the US, the UK, and Russia to my knowledge considered it a useful military skill, though it was usually felt preferable to dismount and shoot from the ground. Training the horse to not shy at the sound of gunfire is probably the most important part of training. It is well documented that some American Indians fired from horseback during attacks, but I am unaware of any studies as to their accuracy. As an example of what is actually possible in this field, do some research on Annie Oakley, and her demonstrations of mounted shooting in the Buffalo Bill's Wild West Shows. It is not something I can do effectively, but some people can shoot amazingly well from horseback.