  This article considers a class of metastable non-reversible diffusion
processes whose invariant measure is a Gibbs measure associated with a Morse
potential. In a companion paper [32], we proved the Eyring-Kramers formula for
the corresponding class of metastable diffusion processes. In this article, we
further develop this result by proving that a suitably time-rescaled metastable
diffusion process converges to a Markov chain on the deepest metastable
valleys. This article is also an extension of [45], which considered the same
problem for metastable reversible diffusion processes. Our proof is based on
the recently developed resolvent approach to metastability.
