  A coupling by reflection of a time-inhomogeneous diffusion process on a
manifold are studied. The condition we assume is a natural time-inhomogeneous
extension of lower Ricci curvature bounds. In particular, it includes the case
of backward Ricci flow. As in time-homogeneous cases, our coupling provides a
gradient estimate of the diffusion semigroup which yields the strong Feller
property. To construct the coupling via discrete approximation, we establish
the convergence in law of geodesic random walks as well as a uniform
non-explosion type estimate.
