I've flown a bunch over the years. I have two rules:
1. I never get involved with a trip that consists of more than 2 legs. First, (with rare exception) I think it's stoopid to move a pet 1/2 to 2/3 of the way across the country, surely it can find a home or shelter closer. And, more importantly, there's way too much that can go wrong on 4...5...6 leg trips and it puts too much pressure on pilots to do something that they may not be comfortable doing.
2. it must be a non-profit at both ends or be going to the adopting party. Preferably the latter, those are the most fun and fulfilling, especially when kids are involved.
Watch out for breeders scamming rides for the dogs they've sold. It happens.