I flew two dogs from Los Angeles to Flagstaff thru Pilots n Paws. It was a rewarding experience..just make sure the dog crate from the rescuer doesn't smell like pee.
Just be aware. There are people on PnP who use these flights, often the multi-leg ones, to transport animals they are calling rescues, but are really animals they have paid a breeder for and are likely selling them across the country for a profit.
There is apparently a whole little cottage industry now of calling animals rescues which have been bred to resale. The claim is the price is just reimbursement for the costs.