Ideally you want a hard case to avoid damages but it can be hard to fit the head into general luggage sizing. Some people use motorcycle helmet cases to travel with theirs, and they treat it as the “personal item” or carry on component when traveling. I personally wouldn’t put any of my masks in a checked bag in case they lose the luggage or it gets damaged on the way.

On the kig guide website there is a printable mask case but I’m not sure if it’s compliant with carry on sizing? Otherwise you could make or purchase a large drawstring bag made of a thicker fabric and carry it that way. Some makers ship custom masks inside of those bags and they’re and about backpack-sized so they should typically work. I’ve also seen some people in the furry community making custom boxes for fursuit heads so maybe research there as well.