u/Randy_Brown recommends using drawer liner from U.S. General for movement, but reports that it's not very good for sound dampening:

I’ve been looking for material to put beneath my typewriter to keep it from sliding around while typing. I realized that I already had something, it’s called nonslip drawer liner that’s used in tool boxes. There are several brands but the one I have is called U.S. General and it was only $7 per 18in X 72in roll. I bought it at Harbor Freight here in California but I’m sure other brands are available at Home Depot, Lowe’s. Ace Hardware, and others stores nationwide.
It’s a thin layer of black (or extremely dark gray) foam-like material that has a nonslip sticky quality to it but not in a Scotch Tape or duct tape adhesive kind of way. I cut it to fit just outside the bottom feet of my Smith Corona Silent Super and it works perfectly.