You've raised a good point that is useful to the discussion, but isn't necessarily a flaw in the study. Your point doesn't change the study conclusion that the current undocumented immigrants in the US are less likely to commit crimes, regardless of the reason. What it does is raise questions whether new undocumented immigrants coming are also less likely to commit crimes. Unfortunately, if recidivism is a large factor, then it would actually support the current policy of making it difficult to obtain citizenship and deporting the criminals.
Label: no hate