Like any other sport, you want to start any specialty training with a BROKE horse. One that stops easily from any speed and travels on a loose rein at any speed, backs up easily and relaxed, knows simple and flying lead changes, responds to leg pressure for side passing and leg yeilds, can do rollbacks or a turn on the fore, neck reins, and everything in between. Basically, you want to be able to control and move any part of the horse's body at any time.
This concept should be the same for ALL horses, no matter what specialty they are primarily going to be used for (reining, barrels, cutting, dressage, jumping, etc). You want to start with a well-trained horse period.
After that, I personally think it is beneficial to do multiple events with the same horse. Do they have to be world champ quality in every event? Of course not. But I think doing something different every now and then (like doing reining for fun with your barrel horse ... Or doing jumping for fun with your dressage horse) can only make your horse more attuned to your cues and keep their mind fresh.
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