Most car rental firms are licensed to do business at Port Everglades and when ships are in port, may provide courtesy shuttle service to/from the cruise terminals to their nearby offsite locations.
Fort Lauderdale-Hollywood International Airport recently opened a Rental Car Center which houses a number of car rental companies servicing both the airport and the seaport.
A single shuttle pickup at Port Everglades is offered by many of the rental car companies at the Center. (they all share the same shuttle bus). A single shuttle pickup at Port Everglades is offered for the following rental car companies at the Center: Alamo, Avis, Budget, Continental, Dollar, Enterprise, EZ Rent-A-Car, Hertz, National, Payless, Royal and Thrifty.
I recommend that you confirm the availablility of shuttle service with your car rental company as contracts between the car rental facilities and the county operated shuttle buses change from time to time.