It's not going really well, is it? It looks like that every political election in the West is a gigantic plea to end or reduce illegal immigration, yet pretty much every study I see about the correlation between illegal immigrants and crime says that yes, illegal immigrants are overrepresented in crime statistics (mainly an effect of them being younger, male and poorer / less educated in average), but somehow crime rates are unaffected by illegal immigration, and never try to explain the "somehow" (the answer is probably that more crimes go unreported and illegal immigrants are not enough yet to tip the global crime scale). The impression is that investigating these topics in too much detail is frown upon (as in: not good for your career) in academia, and it's always more politically prudent to gloss over them.
(see for example: https://pubs.aeaweb.org/doi/pdfplus/10.1257/jep.38.1.181)