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. The necklaces can be a lot of fun for playing night games, and are great on bikes and wagons to help keep other people from running into you. The playa can get very hot during the day, and quite chilly at night. Loose, breathable clothes provide comfort and protection from the sun, and can be layered with warmer things as the sun goes down. Some kids are comfortable with going naked, but need to be protected from sunburn.