In a multi-unit market, a seller brings multiple units of a good and tries to sell them to a set of buyers that have monetary endowments. While a Walrasian equilibrium does not always exist in this model, natural relaxations of the concept that retain its desirable fairness properties do exist. We study the dynamics of (Walrasian) envy-free pricing mechanisms in this environment, showing that for any such pricing mechanism, the best response dynamic starting from truth-telling converges to a pure Nash equilibrium with small loss in revenue and welfare. Moreover, we generalize these bounds to capture all the (reasonable) Nash equilibria for a large class of (monotone) pricing mechanisms. We also identify a natural mechanism, which selects the minimum Walrasian envy-free price, in which for n=2 buyers the best response dynamic converges from any starting profile. We conjecture convergence of the mechanism for any number of buyers and provide simulation results to support our conjecture.