I have friends (U.S. Citizens) who are married to foreign nationals. As part of the application process for an immigrant visa, they are required to produce a clean police record and show financial means or skills for self-support or have a guarantee from their spouse of such support. This are the very basic of a number of logical requirements.
Illegal immigrants, of course, do none of this-- they simply wade a river, climb a fence or pay a coyote. Even during the 1870-1910 era of massive legal immigration, many were turned away at Ellis Island for any number of reasons-- TB, insanity and other health-related ailments being the most common.
And I'm very suspicious about criminal background checks from places like Mexico. Heck, I wouldn't trust LEGAL Mexican immigrants to be who they claim to be. Between the matricula consular and Fox's open collusion with illegals, I wouldn't put it past an applicant to be using a false name, AND to be getting help from Mexican government officials, who don't want any of their invaders to be rejected by our officials.
Which makes it all the more suspicious when they fail to get even a quickie fake police scan check.