The paper has two objectives: proving that the rate of convergence in distribution for mean-field models in CLT regime is $N^{-1/2}$, and obtaining explicit expressions for the infinitesimal generators of two types of measure-valued Markov processes (conditional law of McKean-Vlasov processes, and empirical measures of McKean-Vlasov systems). The proof of the convergence of mean-field system requires the second result about the generators, and both results need to study a notion of differentiability of measure-variable functions known as linear differentiability. Due to the particular framework that is studied, many technical difficulties arise compared to the existing literature. Two of the main problems are the following ones: the CLT regime implies that the limit measure-valued processes are not deterministic, and the empirical measure processes related to McKean-Vlasov equations with jumps are necessarily discontinuous. Both properties make the expressions of the generators more complicated than what is usually considered.