The traditional theater-scenery way is to make the plate long and step
on it to lower the casters and raise the box. Then slip in a brace to
hold it. Could you do it with a hinged toggle linkage that operates by
pulling a rope?

The caster plates I've put under most of my woodworking machines are
hinged on the inside and swing 180 degrees, in to operate and out under
the wall of the base to move. I just lift one end and push the plate in
or out with my foot. The plates have retaining hooks for the
wheels-down position to pass over a door threshold.

jw