Inflation would help an immigrant if they were converting their money from dollars every month. When the dollar rises versus the local currency you have more local currency to spend. Inflation only hurts you if you keep your money illiquid and in local currency.
Yeah, and don't forget, they won't give up citizenship, because they will keep that safety net, despite railing against paying for it. They come back for the last 10-20 years of their life because that "low cost of living" country they are moving to won't be a good deal when they are old.
There's definitely something to be said about the cost of living becoming problematic, but I'm not jumping ship to the US anytime soon. I've seen too many US friends put into catastrophe thanks to health care issues, even when insured.