...SO, you must think about going DOWN with him as more important than going UP with him. You cannot wait for gravity to make you fall back down, you almost bring your pelvis down with the saddle by the contraction of your core muscles...
I don't like disagreeing with tinyliny, but I do disagree here. The only force available to make you go down is gravity. You have nothing above you to push against and force you down. You can grab the front of the saddle and force yourself lower in the saddle, but who wants to ride like that?
Since gravity is the only thing that can make you come down - riding normally - then I draw two conclusions:
1 - The less you go up, the easier it is to come back down. The upward thrust could launch you above the horse's back (did that a couple of months ago, and it hurt when I hit the ground), but the less it launches, the easier it is for gravity to pull you down with the horse's back. So one part of riding the sitting trot smoothly is to go up less.
2 - You can slow your return, but you cannot speed it up. Once you start down by gravity, physics controls how fast you come down - unless you interfere. And the best way to interfere is tension in your legs. Posting without meaning to post. The tension can be your leg muscles straightening your leg against the stirrup, or it can be tension in your thighs pulling your legs together - think of an inverted V coming down on top of an O. Tension pulling the legs of the V together would slow the inverted V from coming down.
A few months ago, I was lunging my mare with an English saddle on her. I noticed the stirrups didn't move much as she trotted in circles around me. But then, stirrups don't fight the horse. I was mostly paying attention to the forward-aft movement, but IIRC, they didn't bounce much up or down either.
So how do you reduce going up? The only things I can think of are to roll your hips back slightly until you feel the fleshy part of your buttocks in contact with the saddle, and let your butt muscles take part of the upward thrust. The second is to have a relaxed back. However, I think there is a limited amount we can do to reduce the upward motion.
To speed up coming down, the only things I can think of are:
1 - Relax, Relax, Relax! Relax your legs. Relax your thighs (pulling together). Relax your butt muscles (particularly while going up). Relax your back. Relax your shoulders. You cannot speed up gravity, but you can slow the effect of gravity down.
2 - Legs below you. I like to ride with a mild chair seat, but if I am pushing my feet forward, then A) there is tension, and B) the weight to my front is pivoting me (creating imbalance) instead of pulling me down.
I will temper this a bit. My legs are extremely tight from 40 years of jogging and only recently taking up riding. The tension in my thighs isn't so much muscular as it is tendons and muscles I cannot control trying to keep my legs together. When my feet are a bit forward, my legs don't have to spread as far apart, so a MILD chair seat might help me. But I'm a 53 year old guy. On the whole, getting my heels underneath should help me come down straighter and faster.
Still, I'm very much a beginner rider. There may be a lot of other things I'm just not seeing. But right now, I am convinced my biggest problem with riding is fighting the horse's motion instead of accepting it.