Being an old guy, I have a classic microeconomics textbook in my library, “Microeconomic Theory: A Mathematical Approach”, by Henderson and Quandt. It is the Second Edition, published in 1971.
Here is a quote concerning microeconomic multimarket equilibrium.
“All the individual, and therefore the aggregate, functions are homogeneous of degree zero in prices. Consumer behavior is determined by exchange ratios rather than absolute prices. . . . . Entrepreneurs’ s excess demands are also homogeneous of degree zero in prices.”
Roughly, this means that if there is microeconomic equilibrium in real terms at one set of prices, then the same real microeconomic equilibrium exists for all sets of prices that are scaled up or down form the initial set of prices.
A wage is the price of labor. Thus, the above perspective implies that if any wage is increased, then the same real microeconomic equilibrium that existed prior to the wage increase is also consistent with all prices being increased by the same percentage. This suggests that the long-run impact of increasing the minimum wage by X% is consistent with a long-run impact of a one-time inflation of X%. If so, there is no reason to expect a long-run benefit from a minimum wage increase for the workers it is supposed to help. In contrast, the short-run impact of a minimum wage increase hurts the lowest wage workers. Minimum wage increase advocates appear to be either uninformed or insufficiently skeptical.