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If you take a US registered vehicle to Mexico, you can drive it temporarily in the Mexico Free or Border Zone as a temporary visitor (FMM), Temporary Resident, Temporary Resident Student, or Mexican citizens living abroad. Everywhere else in Mexico, you have to get a Temporary Vehicle Import Permit for Mexico through the Banjercito website. Your immigration status still applies to the TVIP.
Regardless, you have to purchase Mexico auto insurance from a Mexican insurance company like Mexpro to be able to drive legally anywhere in Mexico. Your US auto insurance only covers you for the first 5-25 miles (dependent on which insurance carrier you use) outside of US territory in Mexico. Some insurance carriers will specifically exclude coverage outside of the country all together.