I appreciate that this an old thread but in the belief that it may be revisited by others in the future I give my experience here. We brought three cats out with us from England with PetAir who were very good, empathetic and professional. I am glad that we kept our cats with us and that they ended their days with us. During our time here we have taken on and neutered (paused while I totted it up) six stray cats/kittens, as well as three dogs, and given them a home. I guess what I am saying is that I totally understand why people want to bring their pets with them but also that there is the possibility of supporting the local strays as well.
We have a good system for cats in our village which has been in place for four plus years in one half and three plus years in the other. The first half has not seen any new born kittens for a few years now and there are significantly less in the other half. Twice a year volunteer vets visit for free medical care and neutering. Volunteers in the village feed the cats twice daily and some even take cats for neutering in between vets coming to the village.
Brought both of ours here and didn't regret it. As for neutering, the vets do a fantastic job. They did 54 cats in one day this week in the Apokoronas. That's down from previous session and hopefully it's finally having an impact,