The elf on the shelf moves using a complex system of gears and pulleys that allow it to move its limbs and head. The toy is designed to be as lifelike as possible, with its limbs able to bend and rotate to mimic human movement. The gears and pulleys are powered by a small motor located inside the elf's body, which is activated when the elf is moved or touched. This allows the elf to move and interact with its surroundings, adding an extra level of realism and playfulness to the toy.