In your specific case and with probe changes, there is no danger because the eddy currents do not create a stray field expansive enough to cross the magnet wire and induce anything. The physical force applied internally to the coil is less than the mass of the falling object. Mostly it just changes the homogeneity. However, the generic answer is yes. If any extra current induced into the SC wire causes the total current to exceed the critical current for superconductivity, the wire will quench. If you forcefully and rapidly move the ring, then the eddy stray field will reach farther. At some point, the eddy stray field will reach far enough to cross a coil, even if only a cryoshim coil, and induce a current change. Whether there is a quench depends on how close to the critical current the wire is already operating. Induced current change is not the only danger from conductors in motion within the magnet field. There is also the physical force. Whatever force is applied to pull the ring away is coupled magnetically to the main coil, so an equal force is pulling directly on the magnet coil, former, and internal mounting structure.