We study the approximation of mean field control problems by systems of interacting controlled particles, focusing on quantitative convergence rates for the associated value functions. We first establish non-asymptotic convergence bounds in discrete time, allowing for common noise and interactions through both states and controls. We then extend these results to path-dependent models, where the convergence rates reflect the increasing dimension of the discretized path space. Our approach relies on dynamic programming arguments combined with Wasserstein estimates for empirical measures of independent, not necessarily identically distributed random variables. In continuous time, we combine these estimates with uniform time-discretization bounds to obtain an explicit logarithmic convergence rate for path-dependent McKean–Vlasov control problems. Numerical experiments illustrate our theoretical results and highlight the influence of path dependence on convergence rates.