We study the heat equation on time-dependent metric measure spaces (being a dynamic forward gradient flow for the energy) and its dual (being a dynamic backward gradient flow for the Boltzmann entropy). Monotonicity estimates for transportation distances and for squared gradients will be shown to be equivalent to the so-called dynamical convexity of the Boltzmann entropy on the Wasserstein space. For time-dependent families of Riemannian manifolds the latter is equivalent to be a super-Ricci flow. This includes all static manifolds of nonnegative Ricci curvature as well as all solutions to the Ricci flow equation.