Here's a little experiment: make a fist with both hands and squeeze as hard as possible. While doing that, try to blow air. How does it feel? Not very good of course.
Now relax both hand and blow. See how natural and comfortable that feels.
We tend to press our valves or pistons too hard when we play. Especially for high or difficult passages. This creates tension in the harm which affects the rest of the body and stiffens the air flow causing problems in the high register.
Take a couple of minutes to practice pressing your valves gently with just the minimum amount of pressure. Trombone players can practice holding the slide gently and moving the harm fluidly. Then play on your instrument this way. Your playing should immediately feel more relaxed and comfortable. You might miss a bit more notes at first but it'll only take a few minutes for your embouchure to get used to this new relaxed way of playing.
This will help your tone in the high register and improve your endurance.
I am associate principal horn of the Singapore Symphony Orchestra and the author of the progressive methods. I'm happy to share my experience as a horn player and teacher with you.