We've got a Schwinn 20" that has good gearing for cruising but not so much for hill climbing. It's a trade off we decided was acceptable. This bike has a very short seat tube and not a super long top tube, so by moving the handle bars back a little, it should accomodate smaller riders. Couldnt tell ya the model but its black and green, BMX style and only a year old. Was $100 at a sporting goods store.
Mongoose used to make decent bikes, both 16" and 20", that had good gearing and medium weight. All of ours were older dumpster finds that I brought back to life so I'm not sure if any of the newer stuff is any good. If you're good with fixing bikes, I suggest this route for growing kids. Replace old/broken stuff with quality parts and you'll have a bike that'll last for years.