In this chapter we explore some aspects of dynamic optimal transport, in particular for the Wasserstein-2 distance on \(\mathbb {R}^d\) , with and without entropic regularization. The unregularized setting is treated in Sect. 2. We put a special emphasis on the primal-dual structure of the dynamic Benamou–Brenier formula and explore how the dynamic dual potential directs the movement of mass particles via its gradient in a drift equation and how this dynamic dual potential is linked to the static Kantorovich dual potentials via the Hopf–Lax formula. We obtain this formula directly via convex duality of suitable intermediate multi-marginal problems.