A kick scooter folds small on its own, so moving it well is mostly about protecting the grip-tape deck from scratching everything nearby, and from being scratched by everything nearby.
Pro tip: A scooter fits neatly into an unused corner of a moving box full of soft items like stuffed animals, using the soft items as padding on both sides.
Watch out: Do not stack heavy boxes on top of a folded scooter; the fold hinge and handlebar clamp are the weakest points and can bend under weight.