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Travel is not kind to printers.....
For ink jet printers, you'd need to at the very least brace the print head so that it can't move, otherwise it can move side to side and smash itself against the inner sides of the printer. But air travel isn't good for ink jets as the reduced atmospheric pressure in flight can suck the liquid ink out of the cartridges and printer innards and cause it to leak in all sorts of places.
For laser printers, getting shaken and jostled can result in toner spilling everywhere and causing a huge mess. But more importantly, there was a bomb threat a few years ago that involved toner cartridges and thus the TSA has treated them in the past like potential bombs. While the ban is officially lifted, it just takes one zealous TSA officer to make a mess of your printer. Not advised.
In your case, since you have an ink jet and are intent on traveling with it, I'd try to brace the printer head from moving with towels (that you're happy to discard if ink gets on them). Take the ink cartridges out and put them in a ziploc bag (as they may leak). And then double-box the printer using plenty of clothing and/or packing peanuts.
Frankly, I'd sell the printer for $50 on craigslist and buy a new one at the destination, especially as a new set of ink cartridges alone costs around $100. Printers these days are sadly disposable commodities....