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

Super-Ricci flows and improved gradient and transport estimates