  We establish a quantitfied overdamped limit for kinetic Vlasov-Fokker-Planck
equations with nonlocal interaction forces. We provide explicit bounds on the
error between solutions of that kinetic equation and the limiting equation,
which is known under the names of aggregation-diffusion equation or
McKean-Vlasov equation. Introducing an intermediate system via a
coarse-graining map, we quantitatively estimate the error between the spatial
densities of the Vlasov-Fokker-Planck equation and the intermediate system in
the Wasserstein distance of order 2. We then derive an
evolution-variational-like inequality for Wasserstein gradient flows which
allows us to quantify the error between the intermediate system and the
corresponding limiting equation. Our strategy only requires weak integrability
of the interaction potentials, thus in particular it includes the quantified
overdamped limit of the kinetic Vlasov-Poisson-Fokker-Planck system to the
aggregation-diffusion equation with either repulsive electrostatic or
attractive gravitational interactions.
