The Mexicans have some strict requirements when you are transitting Mexico from USA to Central America. There is insurance, then there is a deposit you must leave (usually a credit card imprint), and a special sticker. If you didnt get all that in order when entering Mexico then they won’t allow to go beyond Mexico...unless you return to the USA. My new brother-in-law just successfully down his Volvo to Belize within much trouble after complying with all the Mexican rules upon entering Mexico. Seems like a complicated process but he was happy he dotted all the "i"s and crossed all...
Just do your homework before you drive through Mexico. Stop in at an embassy and get an in-transit visa, and follow the rules at the entry border.
I’ve done it several times with no problems. Some people just don’t respect the fact that there are different ways to doing things in different places. Show respect, not arrogance, for the people you deal with along the way and you should have smooth sailing.
If the car came through as a tourist visa to Mexico, there can be a problem. The car has to be entered intransit and there is a procedure for it. Bond deposit which is refunded in Mexico and a couple of minor permits involved.
Ask at the border of Mexico. In Matamoras for Brownsville, everything is done in the same room. Immigration, car bond, car insurance, sticker for window, etc. When you get to Chetumal, you go to a bank and pay for your tourist visa and at the Mexican border, the ADUANA cashiers office, gives you back your bond in USA currency cash. They take the sticker off your windshield. Everything is computerized, so no sense in lying.
If you are carrying goods, make out a manifest list in Spanish and English naming everything you are carrying in the car, destined for Belize. I believe electronics are prohibited. Checkpoints will get a copy. Photocopy about seven to give different people, including the entry point at Matamoras. Solves a lot of problems with the manifest.