They have to run so long that batteries would get used up left and right. The problem with cheap units is that they may not last very long. The idea is that if the child doesn't take to the hobby, you haven't lost much - and if he or she does like it you'll probably want to get a ruggeder machine that can take the pounding of hour after hour of turning with a bunch of loose stones tumbling around in it. Polishing stones takes quite a bit of patience.
Rotary tumblers take weeks to polish a batch of stones - often more than a month.
Vibrating tumblers are much faster, but I don't think you'll find any that are as cheap as starter kits for children. You'd probably want to be pretty sure it's going to be a long term hobby before you invest in one of those. You might try the Rock Trader classified ads here at Bob's Rockhop to see what is being offered - or links to shops that carry tumblers. If you haven't read the Rock Net FAQ and User Tips yet, now would be a good time to do so.