Not knowing what method you are using to move/ship your printer makes solid advice tricky. On the safe side (so to alleviate a potential mess) it would not hurt to remove your ink cartridges and using your original packing is a win! Not many folks keep their original packing for equipment. You would also be best to keep your printer in a horizontal position.
Also to keep in mind if you are not personally transporting your printer, it's best to ask the source you would use as to their requirement regarding ink. Along with that, if you are moving the printer to the USA via air, printer ink that is greater than 16 ounces is outlawed by the Department of Homeland Security. Pay attention to any laws and regulations regarding shipping security to your destination.
From my personal experience moving, I have never removed my ink cartridges as they are HP and "sealed" thus not able to leak. I have never kept the original packaging and my process is to place the printer either in the truck of my car, or on the back seat.
Providing more details about your move will help additional member contributions. Good luck with your move!