When you have applied for temporary residency, you have a specified period of time in which you must enter Mexico, and from there, another period of time during which you must go to Immigration and get your residency finalized. IF you are driving, and are in the first round (year) of getting your temporary residency, you have to get a 180 TIP for the vehicle when you enter the country. The issue is that once you get your residency started, your TIP is no good, and if you stay for over 180 days, you have nothing at all and lose your deposit.
When you drive in again, in order to get a 180 TIP on your vehicle you have to have an FMM. If you get an FMM your temporary residency is no longer valid. If you try to get a 180 day TIP with your temporary residency card, they will not issue one. Catch 22.
Three years ago, friends/neighbors started their temporary residency time. Thy pursued it and pursued it, and after two trips up to Lazaro Cardenas from Zihuatanejo, they were told that all they could do about the vehicle is take their chances on driving the vehicle illegally, and to be sure to get it out within the 180 days or forfeit their deposit. They pursued it through Mexico City and immigration at the airport, and were told the same thing.
Maybe it is possible to legally drive a non Mexico plated vehicle if you have some kind of residency status, but I'd be willing to bet that if something happened, the insurance company would refuse to pay or take care of you.
Not the kind of thing to take a chance with.
As an aside, the regulations for importing a vehicle are very straight forward, and are out there. We moved permanently to Mexico 2.5 years ago. We found that it was not possible to import our collector VW that we had driven for 38 years. Not possible, period. So, we sold it in the US and bought a vehicle here.
Vehicles here are cheap. Depending upon the exchange rate, you can purchase a BRAND NEW Nissan March for $9000 US. Really. The range of used cars is huge. At least in our part of the world, vehicles don't have a large number of miles on them because people don't commute the distances they do in California. Importing a vehicle to Mexico really is not a great option when you can sell your vehicle in your home country and very likely have more than enough cash to purchase something better here!