The idea of those (i.e. like this https://www.youtube.com/watch?v=AWojYBhvfjM&vl=en) is that they're an example of equi-potential behavior.

The superconductor can move because it's staying at a consistent height.

The video linked is also pretty good at showing similar behaviors to the sample of LK99 we see claimed in the video: the superconductor can wobble, shake and does return to it's original position (or tries to) - but it's got a lot more mass.

And when you change the mass of the train the height of levitation changes.

What do you mean "what about?" them? Can you post a video of any of those that don't jiggle when poked?