A study that just came out from the National Bureau of Economic Research provides some evidence to think that even if that process does occur, it's not very significant. It concludes that small firms (as well as other firms) really do take a hit in profits overall when you raise the minimum wage:
We report evidence showing that firm profitability was significantly reduced (and wages significantly raised) by the minimum wage introduction. This emerges from separate analyses of two distinct types of firm level panel data (one on firms in a very low wage sector, UK residential care homes, and a second on firms across all sectors).