USA car in Mexico

As a tourist you can drive a US/Canadian plated vehicle but it must leave Mexico every 6 months to renew the TIP and return. As a Residente Temporal you can drive the same vehicle for the length of the visa - 4 years. After that you must apply for Residente Permanente and the car must leave Mexico.
Passing through or driving in the US with a foreign plated vehicle is not a crime. However after residing in California for 30 days the vehicle must be reregistered and insured. I'm sure there is some wiggle room if you are stopped before registering the vehicle.