We study the empirical approximation dynamics to McKean-Vlasov processes, introduced in Du et al. (Bernoulli, 2023), without restricting ourselves to the weak interaction regime. Our method is composed of two main ingredients. First, using reflection coupling techniques, we show that the empirical approximation is contractive in the Wasserstein-1 sense, leading to the uniqueness and existence of the invariant measure. Second, under the assumption that the original dynamics is a mean field Langevin with finite-dimensional and convex energy, we apply functional methods to control the distance between the invariant measures of the empirical approximation and the original McKean-Vlasov. Finally, by combining the two ingredients above, we propose a discrete annealing scheme which converges to the McKean-Vlasov's invariant measure. This is a work in progress with Kai Du (Fudan), Zhenjie Ren and Florin Siciu (CEREMADE).