Training Science Articles:
As ethical trainers we aim to use positive reinforcement and reduce the use of aversives in our training where we can. This can be difficult when thinking about behaviors we don't want - particularly ones that can be detrimental to the horse themselves or dangerous to the handler. Some behaviors need to be reduced, for their safety or for ours. Luckily we have options! Karen Pryor's book is the origional source of this information in her book "Don't Shoot the Dog" (which I strongly recommend), but I've adapted the information more specifically for horses.
"Teach an incompatible behavior" - This is my favorite, and self explanatory. You just teach a behavior that can't be done at the same time as the one you don't like. Standing instead of bolting. It's easier for the learner to offer you a behavior they know will work, rather than NOT offering a behavior at all.
"Shaping the absence" - Rewarding when the behavior doesn't happen (A horse who bucks every time he's asked for an upward transition, reward the times they don't buck). Behavior that's rewarded increases, behavior that isn't rewarded decreases. This is easy to do once you shift your focus to noticing when they're good, rather than when they're bad. This takes good timing and observation, but it can be pretty hard to adjust our mindset to actually see when the behavior isn't happening.
"Change the Motivation" - Find out what causes the behavior and change the situation (the horse bucks from an ill-fit saddle, so find a saddle that fits). This works really well - if the behavior isn't triggered, it won't happen. It is usually easy to implement. Sometimes it's hard to find what the motivation is to know how to change it. (Is the horse bucking from excitement, or ill-fitting tack, or an unbalanced rider - until you know what causes it you can't fix it)
These are the basic ways to get rid of behaviors we don't like. We all have our preferences - and each behavior may need a different approach. My favorite choices for horses are usually "teach an incompatible behavior" and "change the motivation" those are usually the easiest to implement and the most effective.