Get to know your neighbors. The community is one of the great strengths of Burning Man. These folks will help look out for your kids, your bikes, and everything else at your camp, while you do the same for them. Everyone does, but it's not possible. Most kids don't have the stamina to keep up with adults in a challenging environment, so you'll come out ahead if you help them get around.
Bikes are good for older kids and adults, and wagons or bike trailers are good for younger ones.
Load up a wagon with drinks, snacks, fun stuff, and kids, so you can keep going when they've run out of steam. It can also be a mobile base for nighttime forays, such as going to the Burn. When moving around at night, make sure everyone has lights for themselves and their vehicles, preferably ones that help you keep track of each other. Flashlights are OK, but they're a bit harsh on the playa at night, and break easily if dropped. Glowsticks are much better, particularly multi-colored necklaces and bracelets.