  We study a nonlocal approximation of the Fokker-Planck equation in which we
can estimate the speed of convergence to equilibrium in a way which respects
the local limit of the equation. This uniform estimate cannot be easily
obtained with standard inequalities or entropy methods, but can be obtained
through the use of Harris's theorem, finding interesting links to quantitative
versions of the central limit theorem in probability. As a consequence one can
prove that solutions of this nonlocal approximation converge to solutions of
the usual Fokker-Planck equation uniformly in time -- hence we show the
approximation is asymptotic-preserving in this sense. The associated
equilibrium has some interesting tail and regularity properties, which we also
study.
