I have four sisters, and when we were teenagers, we tried menstrual cups. Consistently none of us could use them, as you say, they move and slip when the woman walks. There must be someone in the world who doesn't have this issue because companies keep making them, but it really doesn't make a lot of sense. The body is so flexible that a rigid cup would slip, and if a cup was that flexible it wouldn't stay anywhere reliably either.
I've personally never tried the cups. The idea of cleaning that doesn't appeal to me. But I can see how it may be more environmentally friendly? Women sleep in these? What happens if "your cup runneth over"?