You MUST show their vaccination record along with the healht certificate. In fact, the last time I flew a small dog in they were more interested in the vaccination record than the health certificate.
Also, it is best/easiest not to bring pet food. Just buy more when you get to the U.S. and vise versa when coming into Mexico.
You must have their vaccination records (up-to-date and they don’t recognize a 3 yr. vaccine) and an international health certificate. Not all vets can give you and “International” health certificate. The vets name and license must appear on the certificate.
Plus, make sure the U.S. Health Certificate states that your pet has been examined AND tested and found to be free of INTERNAL and EXTERNAL parasites.
Be sure to check on your customs form ‘YES’ to live animal.