This is awesome info. How far out of the area are you willing to go with hiring? Say if someone was hiring for a position helping immigrants in general with visas and an applicant had worked only with asylees instead. Too dissimilar or close enough? Not my specific experience, but just as an example. Also, do you have a general cut off for experience (number of months/years?)
In my experience we don't. I work in Manhattan, so we don't have any reason to look beyond the law schools in this area (though contrary to popular belief on these forums, while school name is important, e.g. you going to NYLS and ranking in the bottom 50% is going to catch my eye as a "meh" candidate, we don't automatically go with the NYU/CLS kids either).
Your example is fitting since one of the things we do is file U-Visas, so I can say what my thought process would be because I've had this happen before. I'd want to know a few things: how many years of experience doing asylum work? How many cases? Did they work with asylees from only one or two regions of the world or many? Was this part of a FT job or just an internship? Does this person speak other languages? Does this person articulate a direct link between their prior experience and what we specifically do? Long story short, I wouldn't discount that resume right away, but you can see how already there would be some questions that I'd want answered. It's difficult to say what a "cut off" would be in terms of experience, but I can say that if one candidate had the work experience as part of a FT job either before/after law school (tends to be at least a few years, mine will be 5+ years prior to law school) and the other got their experience through an internship/externship (less than a year in 99% of cases), I'd probably lean toward the former initially unless there was something spectacular that the latter did. Ideally I would get the answers to my initial questions from the resume, and if they look positive, I'd probably give that person a first-round interview and see how they fare from there.
EDIT: Oh, this kind of work is definitely not for everyone. Crazy things happen, but I personally enjoy the fact that my job is never boring (even if my clients sometimes makes me want to throw my computer out the window...marathon and half marathon running is a fantastic stress relief). My mother was a detective while I was growing up, so I'm used to the crazier side of life. Some people aren't, and they wash out of this kind of work after one or two years.