No, Mexico will have no record of your conviction and have very few computers to check up on you.
But you do realize ... you just ride into Mexico ... you do not have to stop?
If going beyond Sonora on the mainland, you will need a TVIP (vehicle import permit). You also need a tourist visa (buy at border if spending more than a few days)
If you are wanted you will get arrested at the US border when you are coming back North, entering the USA. They will run your plate and your passport. Instantly they know your status. A DUI should not affect anything there unless you have an outstanding warrant.
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