Well.. I managed to bring my 27 imac and my wife's 21.5 imac in their boxes to the cabin free of charge thanks to Turkish Airlines for a few times. It fits inside the overhead bins and the airplanes were Airbus A320, A321 and Boeing 737-800.
I use an LTA iLugger for my 21.5”. I just wear it as a backpack, but you have to ensure you specify the backpack straps as they were no longer included by default when I ordered my last one.
I keep the box for my 27” iMac and it served me well for several moves to different houses and even went in a shipping container for a transatlantic journey once - no issues.
Another option:
Sell the machine and buy the same machine or any better one second hand, or new, after you moved.
Served me well for many items, including computers.
Alternatives:
- Mac Mini (just pack it in your luggage -- get a display "wherever you're going")
- MacBook Pro (again, add keyboard, mouse display "when you get there")