  We introduce the notions of `super-Ricci flows' and `Ricci flows' for
time-dependent families of metric measure spaces $(X,d_t,m_t)_{t\in I}$. The
former property is proven to be stable under suitable space-time versions of
mGH-convergence. Uniformly bounded families of super-Ricci flows are compact.
In the spirit of the synthetic lower Ricci bounds of Lott-Sturm-Villani for
static metric measure spaces, the defining property for super-Ricci flows is
the `dynamic convexity' of the Boltzmann entropy ${\mathrm Ent}(.|m_t)$
regarded as a functions on the time-dependent geodesic space $({\mathcal
P}(X),W_t)_{t\in I}$. For Ricci flows, in addition a nearly dynamic concavity
of the Boltzmann entropy is requested.
  Alternatively, super-Ricci flows will be studied in the framework of the
$\Gamma$-calculus of Bakry-\'Emery-Ledoux and equivalence to gradient estimates
will be derived.
  For both notions of super-Ricci flows, also enforced versions involving an
`upper dimension bound' $N$ will be presented.

Differential Geometry;Functional Analysis;Metric Geometry