A quick thought with no actual data to back it up:
We presently believe that people who are long-term unemployed are in crisis — that their resumes are filtered out of possible reemployment in an early round, before they can get a chance to interview or otherwise demonstrate their worth.
There are also people who believe that an increase in minimum wage is an effective tool of long-term economic growth, and that the disemployment effects of the minimum wage are minimal, or are counteracted by an overall healthier labor market.
It seems to me that even if a minimum wage increase leads to a stronger labor market over time, if it has even short-term disemployment effects, and we believe that being unemployed for more than a few months leads to long-term employability problems, that a minimum wage increase could create a polarization of the low-end labor market — helping some people while rendering others permanently unemployable. And that we should be particularly sensitive to the idea that even if someone has a shitty minimum-wage job right now, there is value in their having that employment history that should not be casually thrown away.
I emphasize that I have absolutely no data backing this up. If anyone out there knows of studies which speak to this idea, I’d be really interested in seeing them.