  By means of a space-time Wasserstein control, we show the monotonicity of the
W-entropy functional in time along heat flows on possibly singular metric
measure spaces with non-negative Ricci curvature and a finite upper bound of
dimension in an appropriate sense. The associated rigidity result on the rate
of dissipation of the W-entropy is also proved. These extend known results even
on weighted Riemannian manifolds in some respects. In addition, we reveal that
some singular spaces will exhibit the rigidity models while only the Euclidean
space does in the class of smooth weighted Riemannian manifolds.
