I can't help much with the jumping part... normally when a horse changes leads like that it is because they are thinking about going in a new direction. Maybe he is getting intimidated by the jump and thinking about ducking out? Idk, I don't know anything about jumping lol
For lead changes, you have GOT to have control of your horse's hip. You need 1. Forward motion, 2. Lateral movement, 3. Collection, and 4. Elevation. I would start with getting him reallyy good at hipping in and two tracking. Start at the walk and progress to the lope. Once you can two track well at the lope, here is how I get a lead change:
One method is a figure 8 method. Begin by establishing a circle in one direction (lets say left for the sake of explaining your hands and legs.) Go around as many times as you need to to get your horse nice and calm and collected. Then when he's settled, as you come across the center where you would change directions, simply ask your horse to two track towards the center of your circle by lifting up on your right rein and putting your right leg into him. If he doesn't respond, or is leaning against your leg cue calmly stop him and side pass him to the left, ensuring that he realizes that pressure means to MOVE! Then pick up the lope and try again. I would go through about 3 times getting good leg yields in the center. When you decide to change, two track to the center (right rein up and right leg) and when you are ready to change, simultaneously open up your right leg and apply your left leg back behind the girth to encourage his hip to move back over to the right. MAKE SURE you do not drop your right rein! You must keep that rein up to keep his shoulder elevated! If you drop the rein, you'll drop his shoulder and he won't be able to change. You also need to maintain a decent speed. Mechanically, this is the best way to get a lead change.
There is another method involving a counter canter which is essentially the same as above. Get on a large circle with a counter canter and as you come out of the middle of the arena and approach the fence, two track him towards it (if going to the left again on the right lead, left hand and left leg). Do that several times until he gets the idea, then do the same thing as above. Two track right using left hand and left leg, then when you decide to change, keep your left rein up so that the nose stays tipped to the inside of the circle, then switch your legs asking him to move his hip to the left. If you go more at an angle into the fence, this can help encourage them to change... This is also good for difficult horses because counter cantering is uncomfortable, making them want to be on the correct lead.
I hope I described that well enough...
Thanks and Gig'em!