Be sure to check with the airline you are flying with to confirm their rules for flying with a firearm.
If there was a problem in one of the anti firearm locations (your layover stops) I would make sure the airline maintained control of the luggage containing the firearm as to avoid any possible criminal charges.
Personally, I would prefer to ship my firearm/s via UPS/FedEx next day service to the address I was staying at.
Most/many hotels allow this service for newly arriving guest, and well family is family.
Q: May a nonlicensee ship firearms interstate for his or her use in hunting or other lawful activity?
Yes. A person may ship a firearm to himself or herself in care of another person in the State where he or she intends to hunt or engage in any other lawful activity. The package should be addressed to the owner. Persons other than the owner should not open the package and take possession of the firearm.