Another case for The Fringe team.
My Guesses:
1. Materials expanding and contracting at uneven rates as a result of temperature differences in the metal and small upward thermals created by ground moisture evaporating. The thermals are warmer nearer the ground and as they rise it varies the temps along the chains of the swing vs the temp of the surrounding air causing them to be slightly different in temps along their length thus expanding and contracting at a different rate along the length of the chains. The chains holding the swings are also not exactly the same length which accounts for the twisting and sidewards motion (which implies that either a mistake was made in measurement or they were set that way deliberately for this trick). I'll bet if they replaced the chains with rope you would not see this happening. The shape of the seat also helps as it already provides maximum air current resistance from the top and bottom, which means different air density at the tops and bottom of the seat, such a difference causes lift which adds to the initial movement to get things going. The denser moist air from evaporating ground water creates thermals that hits the bottom of the seat and the lighter air at the top of the seat helps induce a small amount of lift, the swing starts moving, and the differences in temp along the chain keeps things going. Its the long train effect; why very long trains need an initial jerk to start the whole thing moving - the seat lift is the jerk and the expansion and contraction is the power that keeps things moving. The time limits it gives of 5 to 10 days also coincides with water evaporation actions in cooler/colder weather as you notice people are wearing coats in some of the videos. Water evaporates over a longer period of time in cooler/colder weather but it still evaporates, the sun hits it or the air warms and small thermals are created. I'd be willing to bet that this doesn't happen at night, you notice that its daylight when this happens, thus accounting for warmer air or sunlight activity to produce the thermals.
or....
2. Vibration from nearby traffic.
or...
3. The result of the exploding star in the Q3 quadrant near the crab nebulae.
or.....
4. It could be the ball bearings, yep, thats it, the ball bearings. Who woulda thunk.
or....
5. Its the internet, what did you expect.
(number 1 is my best guess - i'm not saying its correct mostly in case i'm wrong

its just a guess thats all, besides it does sound pretty good and explains everything

)