Several places on this web site I've discussed how to teach horse and rider Half Halts (HH aka almost halt but at last second continue on in current gait). Use them every 5-6 strides (ONLY works if done CORRECTLY as it's not just reins) to rock horse back on it's butt.
So in jumping, start with smaller jumps since larger jumps can intimidate a horse and mentally spin them up and get them rushing.
Then use the HH. As soon as you canter so a HH every 6 strides, before every corner, 4 times within a circle and start with at least 2 times BEFORE the jump and within 4 strides AFTER the jump. If at any time you do not feel the horse hestiate like they are ready to stop then halt horse immediately and make it stand ( my mare HATES standing hence when she's "punished" by having to stand SQUARELY and quietly she less likely to do her bad behavior - which for her is rushing, flying changes every stride or so, etc.).
Some days (especially in the beginning) horse will not have enough mind left to go back to jumping, so turn lesson into disciplined work consisting of listening to everything rider is asking horse to do and responding to it.
I believe the ruhing is a combination of one or both - either he's lost his mind - scared, excited, and/or he's unbalanced. When unbalanced, a horse tends to rush because they're falling on their forehand.
Dressage is for Trainers!