Hi, Jane!
I have a Wavemaster bag, which is also what I've seen in a local karate dojo. You can fill the base with sand or water (I chose water (with a bit of bleach in), since I could figure out how to do that with a hose, it was handy, and if I ever moved and had to lighten it, I can easily tip it on its side on my front porch and let the water run out).
The bag does move somewhat (perhaps less when filled with sand). The only way it wouldn't is if you don't hit hard. I think you have to give up on the idea of it not moving at all.
I used to set mine on a square of that rubber material you put under rugs to keep them from slipping (but couldn't find it when I recently worked out). I've also set up dumbells around the base on the side away from me.
The best solution to minimize movement, IMO, is to combine the rug mat with weights. Another solution is to not stay in one place and move around the bag, so if your punches/kicks move it one way, you can punch/kick to move it back.
It's possible the type of heavy bag Amy Bento uses in her new workout (which has kind of a spring mechanism between the base and the upper part) would move less with this set-up--though Amy's bag moves in her workout, and she has done this.
(another cool, IMO, option is to buy a 'person-shaped' bag : it has the same Wavemaster-type base that you fill, but the top is like a mannequin that you can hit. It can make the workout seem more authentic. Though it is also more expensive than the regular Wavemaster bags.)
HTH!