In a basic flow routine in gymnastics you might see this from level four all the way up to elite. This is called a handstand forward roll, and it is exactly as it sounds. The student will kick up to a handstand after practicing that, perfecting that, and then slowly kind of lean forward as if they are falling flat, and then curl into a ball. It's very important that you don't stop and drop straight down; you want to make sure your momentum still goes forward as you'll drop straight down, you'll hit your head, crunch up and hurt your back. So there are some drills you can do that would train you to hit the handstand, and then continue moving forward in a forward motion and then roll out in that position. This is also done with straight arms or bent arms. It's also done in a either tuck position or straddle position, or a pike. Mainly it would be done in a tuck position, so handstand, forward tuck roll.
Something to keep in mind when you are doing your handstand forward roll is to make sure you stop all the way up in the handstand, no arching, no piking. When you are doing anything involving a handstand, it is very important that you do show that handstand. That is what we are looking for, and that is what makes sense and that is easier to do and is safer to do. So whenever you do a handstand forward roll, make sure you kick all the way up, practicing the handstand, and adding on to that skill of forward roll. And always make sure you know what you are doing, before you do it.