The aim of the paper is to prove that the rates of convergence in distribution for $N$-particle mean-field models are the expected one: $N^{-1}$ in Law of Large Numbers regime, and $N^{-1/2}$ in Central Limit Theorem regime. These proofs require to study empirical measures of McKean-Vlasov particle systems, and conditional laws of McKean-Vlasov processes, as measure-valued Markov processes. In particular, the expressions of the infinitesimal generators of such processes are established for measure-valued processes with general jumps. The generators being differential operators, all the proofs rely on the analytical properties of measure-variable functions, and the differentiation of such functions.