I'm not sure if Google would yield any concrete answers, just you should try. Or contact a vet or livestock-related lawyer to be 100% sure.
I would imagine a negative Coggins and possibly a clean bill of health from a vet would be required for interstate travel, but, again, contact someone with that kind of expertise. Don't quote me on this, but I think a Coggins can be considered a valid proof of ownership. Something else to ask.
ETA: Found an answer with Google.
"Equines (horses, donkeys, zebra & mules): Must be free of contagious diseases. Equines entering the state require a CVI (Health Certificate) and a negative Equine Infectious Anemia (EIA) test within six (6) months of entry. Foals under 6 months are exempt from the negative EIA test only if they are traveling with their negative mother. A California entry permit is not required for equines to enter the state." http://www.cdfa.ca.gov/ahfss/Animal_...html#Livestock