Why is the cage nut spinning or moving when I try to start the screw?

A little movement is normal because the inner nut floats for alignment. If it is moving excessively, make sure the cage is fully seated and the equipment ear is aligned with the square hole before applying pressure to the screw.

Start the screw by hand. If you attack a misaligned floating nut with a powered driver, the nut is likely to win the argument in a rather destructive fashion.