  In this paper we consider compact, Riemannian manifolds $M_1, M_2$ each
equipped with a one-parameter family of metrics $g_1(t), g_2(t)$ satisfying the
Ricci flow equation. Motivated by a characterization of the super Ricci flow
developed by McCann-Topping, we introduce the notion of a super Ricci flow for
a family of distance metrics defined on the disjoint union $\MM$. In
particular, we show such a super Ricci flow property holds provided the
distance function between points in $M_1$ and $M_2$ evolves by the heat
equation. We also discuss possible applications and examples.
