Fallen Heroes Family Travel Act of 2019 This bill addresses the transportation of the remains of members of the Armed Forces who die outside of the United States. It requires the Department of Defense (DOD) to provide, in the case of such a member whose remains are returned through the mortuary facility at Dover Air Force Base, Delaware, transportation of such remains to the applicable escorted remains destination. (Under current law, this service is limited to the remains of those who die in a combat theater of operations.) The bill also requires DOD to extend travel privileges to family members of those personnel whose remains are returned in this manner.