  We illustrate some novel contraction and regularizing properties of the Heat
flow in metric-measure spaces that emphasize an interplay between
Hellinger-Kakutani, Kantorovich-Wasserstein and Hellinger-Kantorvich distances.
Contraction properties of Hellinger-Kakutani distances and general Csisz\'ar
divergences hold in arbitrary metric-measure spaces and do not require
assumptions on the linearity of the flow. When weaker transport distances are
involved, we will show that contraction and regularizing effects rely on the
dual formulations of the distances and are strictly related to lower Ricci
curvature bounds in the setting of $RCD(K,\infty)$ metric measure spaces.
