If you've ever tried to sit in a swing that was completely static, not even moving a mm, and tried to start it swinging with nothing but leg kicks, then you know it's a long process.
That's why most kids get a push, or push off with their feet on the ground to get it moving first.
IMO, all that's really going on is the kid grabs the chain or rope a couple feet above the seat. Holds that portion and what's above it in place, then kicks his feet out until his body is parallel to the ground or nearly upside down, this makes the portion of the chain below his hands start swinging a little. When you relax that grip, the whole chain/rope swings back rather than just the portion below your hands, then repeat that step and the swinging momentum gets larger and larger with each leg kick.
Cool question thuogh.