  Inspired by the recent theory of Entropy-Transport problems and by the
$\mathbf{D}$-distance of Sturm on normalised metric measure spaces, we define a
new class of complete and separable distances between metric measure spaces of
possibly different total mass.
  We provide several explicit examples of such distances, where a prominent
role is played by a geodesic metric based on the Hellinger-Kantorovich
distance. Moreover, we discuss some limiting cases of the theory, recovering
the "pure transport" $\mathbf{D}$-distance and introducing a new class of "pure
entropic" distances.
  We also study in detail the topology induced by such Entropy-Transport
metrics, showing some compactness and stability results for metric measure
spaces satisfying Ricci curvature lower bounds in a synthetic sense.
