Hmm. If it's sliding around, you may need a larger size. I use the large size (got it after a catastrophic Keeper failure with the smaller-sized Keeper) and I haven't had kids.
The cup is supposed to develop suction. If you just pull straight down on the cup without pinching the bottom to let air in, the cup is supposed to more-or-less stay put. (Or, at least, that's my experience.) If you're not get getting a good suction seal, that may mean the cup is too small.
On the flip side, are you sure the thing is fully open? When I got the larger size, I had problems getting it to fully open at first. You want to rotate it to make sure it's fully unfurled, and I usually tug on it a tiny bit to make sure I've got a good suction seal.
I have to say, though, that contrary to the website I've never gotten the thing to be TOTALLY leak-free. I generally get a little leakage on my heavy days, when the cup fills up. Also, unlike a tampon, if your period has already started when you put the cup in, there'll still be a little blood that was below the cup when you put it in, if that makes sense. Like I said, I generally use cloth pantiliners on heavy days.
Still miles better then tampons as far as I'm concerned.
EDIT: Jessibean, I think I have a fairly low cervix, too. At least, contrary to the instructions on the Diva Cup, the Keeper, and every other cup out there, the top of the cup is right by my cervix... and the bottom is right by the entrance, so it can't possibly go any further down! I have no problem with the Diva Cup, but I didn't have problems with tampons, either. (Then again, I used OB, which is a pretty short, fat tampon. I never liked the long, skinny brands.)
Once upon a time these cups used to offer a money-back guarentee. I see from the website that Diva Cup no longer does, but some of the other cups still might.
It might be worth a try. Cups are GREAT for light flow days, since they don't suck you dry the way tampons do.