I've seen this answer, which clarifies that superconductors float on top of permanent magnets not because of the Meissner effect, but because of flux pinning. The superconductor is cooled, and in that process discrete flux centers are created (thin lines of non-superconducting regions). Sometimes a hand-wavy explanation is provided which sound like "and therefore it cannot move because that would change the flux through the flux centers."
But it remains unclear to me why flux pinning prevents the superconductor from moving relative to the magnet. If the superconductor was moved far from the original magnet it cooled near, the superconductor would just make a current that preserves the flux through each of the flux centers. This is more in line with my experience with superconductors. I worked in a lab with superconducting magnets, and the primary observation was that when you tried to change the flux through a closed superconducting loop (like by turning off another magnet), the current in the superconductor would change to preserve the flux. Presumably this occurred only because our superconductors were held in place very strongly.
It seems this kind of argument can explain why it is possible to move a superconductor with a sufficient external force... and somehow gravity is typically not sufficient. In this video the person grabs the superconductor and moves it, and it stays in the new position. (By the way, it's not clear to me why the superconductor in the video can spin, but I'm not asking that right now.)
So what are the details of this argument? What prevents superconducting material from moving when it forms near an external magnet? What happens to the superconductor when you force it to move with a sufficient external force, and what sets the scale of that sufficient external force.
