To bring dogs and cats into Mexico via an airline please check your airline’s website – rules vary quite a bit. Pets are generally not permitted to fly, other than in the cabin (smaller dogs and cats), during the hotter summer months. There are also embargoes on flying pets on many airlines during the Christmas season — actually for weeks leading up to it. The theory, as far as I can tell, is that the cages take up too much room during a busy season with lots of extra suitcases.
When bringing a dog or cat by car, health certificates are no longer required.
Please read here regarding dogs, cats, birds, ferrets, etc. https://www.aphis.usda.gov/aphis/pet-travel/by-country/pettravel-mexico