[en] Let Lt:=Δt+ZtLt:=Δt+Zt for a C1,1C1,1-vector field Z on a differential manifold M possibly with a boundary ∂M , where ΔtΔt is the Laplacian operator induced by a time dependent metric gtgt differentiable in t∈[0,Tc)t∈[0,Tc). In this article, by constructing suitable coupling, transportation-cost inequalities on the path space of the (reflecting if ∂M≠∅∂M≠∅) diffusion process generated by LtLt are proved to be equivalent to a new curvature lower bound condition and the convexity of the geometric flow (i.e., the boundary keeps convex). Some of them are further extended to non-convex flows by using conformal changes of the flows. As an application, these results are applied to the Ricci flow with the umbilic boundary.