I have been involved with many rescues and many attempts to schedule a rescue, and maybe this is as good a place as any to discuss some of what goes on attempting to coordinate one.
When a pilot is first getting involved with a transport it is important to understand that as a pilot your train of thought is not going to be the same as the folks involved with rescues unless they have been involved in aviation. As a pilot I have learned to make every attempt to precede rescues with some sharing of information. I try to help rescues understand my limitations with respect to the available space, and weight and balance considerations. Range and fuel prices are also discussed. I try to help rescues understand the "big" airport is not necessarily the most convenient for them or for me. I often have to help rescues and shelters understand the animals are not going to board a the terminal building, but through the FBO. I try to look ahead at weather and keep informing the rescues involved how the weather is shaping up and if it may adversely affect our planned date for the flight. I am instrument rated and I will fly almost all the time so my advice to VFR pilots and rescues relying on a VFR pilot to be very understanding of one another's needs and concerns. If as a pilot you have limits on when you can fly make everybody aware of them, and do not feel pressured to fly.
As a pilot I would urge you to understand a planned rescue can fall apart at the last minute for more reasons than can be explained here. Animals get adopted, they get ill, the paperwork from the vet fails to become available, or folks at the receiving end cannot take the animals or the dog ate the homework. Stuff happens.
The best advice for both pilots and rescues is to keep the faith. As pilots and rescues develop relationships, get to know one another, establish lines of communication, and work together each transport requires less and less effort because everybody learns how they need to relate to one another and you not only begin to mesh as a team, but you will also find the FBOs you deal with and even ATC knows you and your needs on the rescue flights.
The most critical part for all to understand is that everybody involved from start to finish must be available via phone. If a pilot cannot communicate with the parties sending and receiving the animals anytime it places a huge burden on the pilot. As a pilot I need to know I have reliable parties who will meet me with the animals as planned, and that at the receiving end I will also be met when I land. I absolutely need to be able to communicate if I should have to land unexpectedly so I can have assistance if needed.