We consider Fokker–Planck equations with general nonconvex potentials that have several local minima. In the limit of vanishing diffusion, matched asymptotic expansions predict simple reaction–diffusion dynamics for the masses concentrated in the local minima. We present here a rigorous derivation based on the elementary approach developed in Herrmann and Niethammer (2020) for the case of tilted periodic potentials. The proof relies on suitably defined substitute masses and error bounds that make use of an energy-dissipation relation.