I had a couple from Germany that walked with two dogs, and stopped by the Camino Travel Center to see how they could travel home with them. It seemed that these were rather large dogs (I did not see them), about 30kg each they said.
I called ALSA (the bus) and they would not allow dogs on international trips. They had talked to RENFE (the train) and it seemed that they also had some rule against dogs.
I am not sure what they did to get home. Renting a car would have been one option, but it seemed like they did not have a drivers license....
Anyway, just and FYI that you need to plan how you would be going home with your dog. I am sure it can be done since I seemed to hear about pilgrims walking with dogs all the time, but I am not sure how they travel with them home afterward.