Missing up or down can be caused by an unstable glove, tilting the shoulders back and throwing uphill, , etc. If you are right-handed, then the up-and-in pitches are often caused by opening up the shoulders too early and letting the arm drag through the zone. When you over-correct, the pitches sail down and away.
Definitely open the hips into landing and have the shoulders rotate upon landing. As the shoulders rotate, the chest is thrust out causing an arch in the lower back. I wouldn’t force this arch though. Think of the chest and then unload the back arch, catapulting the upper body forward to release.
Another common cause of missing high is the elbow being low coming through as the shoulders turn.
The hips have to open before the shoulders. If you look at the best pitchers, in many cases their hips leads their shoulders by up to 90 degrees.
This has nothing to do with rushing.
When you are occasionally rushing (e.g. sometimes missing high – especially when trying to throw hard), then your timing is different so that sometimes your foot plants sooner than others. If you are habitually rushing (e.g. always missing high), then you need to so something like break your hands sooner so that your arm is up and ready at the moment that your shoulders start to rotate.