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 Savaré 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.