  We show that the heat flow on super-Ricci flows in the sense of Sturm
satisfies transport estimates with respect to every $L^p$-Kantorovich distance,
$p\in[1,\infty]$. As an application we construct Brownian motions on
time-dependent metric measure spaces and present transport estimates for their
trajectories.
  The proof is inspired by the approach from Savar\'e and Bakry respectively
and takes advantage of the self-improvement of the gradient estimates. For this
we prove a refined version of Bochner's inequality under strengthened
assumptions on the metric.
