Make sure your mover offers these specialty cartons on either a rental or sale basis. If rented, which is cheaper, they can be supplied for you to pack either prior or on the day of the move.
Under a rental agreement, you will be responsible to pack and unpack the cartons that must be returned after the completion of the move.
Make sure the mover is able to give you these cartons at the start of the unloading procedure at destination residence. If he did not place them in an accessible part of the truck, you may not have amble time to complete your task prior to them leaving. If this is the case, bring it to the attention of the Crew Chief who should make some type of arrangement to compensate for this mishap.