Upvote:3
I recently entered Mexico from the US by road at Tecate. Mexican Immigration wasn't present at the crossing, which was completely un-staffed. Leaving Mexico at the same crossing a few days later was the same.
While Mexican Immigration does look at my FMM and passport when I enter by air, and the airline collects the FMM on my way out, the border is not necessarily a fully-sealed boundary.
Thus, the question "When do they do the background check?" assumes a fact not always in evidence.