I just flew my first dog-transport trip, and can share some of my concerns, as a non-dog-owner and pilot of a small airplane:
I worried, would the dog be clean, flea-free, calm? Would he get airsick? Would he need tranquilizers or other meds? My airplane's too small for a big dog in a crate. How would I confine him for his own safety and comfort, as well as for ours?
Fortunately, the people in charge of our dog-passenger really knew their stuff. The web post clearly specified origin and destination, with flexibility for each. He arrived having eaten only very lightly that morning, having been walked plenty, so his bladder wasn't full. He had a leash and a snug harness which was easy to thread a seatbelt through, and he was freshly washed. There would be no need for medication, as, they reassured me, "He rides well." They even provided a blanket to go between him and the upholstery. Perfect! The dog handled the trip with quiet aplomb, his manners were fine, he didn't seem to mind the occasional turbulence or being at 11,000 feet. It was a very reassuring first-timer's experience, and if all dog-rescuers provide their volunteer pilots with such a well-prepared animal, you'll have more of us than you know what to do with.
