Problems that come up at a walk and trot, won’t go away when you’re going faster.
When I (re)start horses (mustangs, colts, or horses who have not been handled in a friendly way), I focus on tension and misunderstandings while the horse is standing or walking. When I find tension there – if the horse is pulling on the lead rope or reins, or not “listening” – then I will not ask him to trot, let alone canter. Because it will come up in those gates as well, except with speed and more power behind it. It makes the whole thing worse for you and the horse.
I don’t see the point in going faster. I work until we are as light as can be at the walk and then slowly move to trot and slowly to a canter… this moves you along faster in the development, because you are not trying to solve this problem in all three or four gates – just in one. I find that if I take the time to get my idea across to the horse, it isn’t much of an issue when I later ask him to do it while going faster.
On a horse’s first ride: if I can’t stop, back him up, or ask him to yield at a walk, he definitely won’t do it at a trot or canter. So take it slow and work through the sticky spots before rushing on – they grow bigger when unattended to…