The empirical measure flow of a McKean-Vlasov $n$-particle system with common noise is a measure-valued process whose law solves an associated martingale problem. We obtain a stability result for the sequence of martingale problems: all narrow cluster points of the sequence of laws solve the formally limiting martingale problem. Through the solution of the limiting problem, we are able to characterize the dynamics of limits of the empirical measure flows. A major new aspect of our result is that it requires rather weak regularity assumptions for the coefficients, for instance a form of local continuity of the drift in the measure argument and ellipticity of the diffusion coefficient for the idiosyncratic noise. In fact, the formally limiting martingale problem may fail to have any solution if there are discontinuities in the measure argument, so that the stability property is in general not true under low regularity assumptions. Our novel approach leverages an emergence of regularity property for cluster points of the empirical measure flow. This provides us with a priori analytic regularity estimates which we use to compensate for the low regularity of the drift.