I looked through lots of books about gears and mechanical movements, and I learned about the Geneva Wheel. That's the secret to everything! I guess it was invented by Swiss clock makers to have intermittent movements instead of a constant and steady motion mocking the movement of the crank.
A Geneva Wheel consists of 2 wheels actually. One has notches in it, and the other has a pin to fit into the notches. As the PIN wheel turns it slides into a notch in the notched wheel. It turns just enough to align the next notch up into the path of the pin, so as it comes back around it repeats all over. You can put as many notches in as you wish... as long as you get them in the right place.
Here's a picture of the various pieces before I put it all together. Depending on the task of the piece, some needed to be plywood for strength. Normally I don't like the look of plywood but when there's danger of fragile parts crumbling there's nothing better wood-wise.
Here's a side view of the inner workings of it all. After I got this thing built, almost immediately after I glued in the last piece, I thought of a way to make the box half as tall. The more streamlined I can get it, the better. I guess my next one will be better. I've also thought of a way to not have that big circular opening in the top of the box.