No connection what so ever. Once Aduana issues you your car permit, the hologram goes on the windshield, you are good to go for 180 days. Stickers are irrelevant. The only condition is the car permit is only good if your immigration status is current and valid. Stories about someone going to jail for an expired license plate tag are pure BULL loney. Yeah a cop "could" make an imaginary issue out of seeing an expired tag. I had one in Zacatecas (a city transito) try and make a fuss over a four year old expired tag. I laughed at him and pointed to the windshield hologram. Wanna know what he did? He shrugged then walked off. He gave it his best and was ridiculed for it. People who don't bother learning basic Mexican law are the ones who fall for this kind of nonsense. I'll bet that idiot cop didn't think it was so funny when the feds came down on his boss because I turned him in via a Green Angels complaint. Got it all: Name, badge number, date, hour, location, the whole enchilada.
The Mexican vehicle permit is another thing altogether. Aduana is nightmarishly strict about being processed at the border and surrendering the permit. Even if you have an accident or get sick and document it, prepare to lose your deposit. The 'official' take of Aduana is "You should have returned here earlier"
Most folks do not know that the 180 day car permit is only good as long as your FMM tourist card is valid. The 10-year RV permit is another thing altogether.