  We introduce notions of dynamic gradient flows on time-dependent metric
spaces as well as on time-dependent Hilbert spaces. We prove existence of
solutions for a class of time dependent energy functionals in both settings. In
particular we are interested in the case when the underlying spaces are metric
measure spaces and the energy functional is given by the time-dependent
Boltzmann entropy or the time-dependent Cheeger's energy. Under the assumption
that each static space satisfies a lower Ricci curvature bound, we prove
existence and uniqueness of the entropy gradient flow and we identify it with
the forward adjoint heat flow introduced in "Super-Ricci Flows for Metric
Measure Spaces" by Kopfer/Sturm. We identify the gradient flow for the
time-dependent Cheeger's energy with the heat flow introduced by Kopfer/Sturm.
