Start with a purchased plastic two-shelf storage cube that's large enough to hold short stacks of clear plastic shoeboxes. Purchase the number of shoeboxes that your cube can hold. (You can position multiple cubes side-by-side if you have a lot of shoes.)
Cut a laminate board base just a few inches larger than the base of the cube. Use white laminate tape to finish the cut edges of the board.
Flip the board base over and screw casters to each corner. Secure the wheeled board to the base of the cube, screwing down through the inside of the cube and into the top of the board.
Place pairs of shoes inside each shoebox and stack the boxes as shown. You're ready to roll!