  We investigate how mass transports that optimize the inner product cost
-considered by Y. Brenier- propagate in time along a given Lagrangian. In the
deterministic case, we consider transports that maximize and minimize the
following "ballistic" cost functional on phase space $M^*\times M$, \[ b_T(v,
x):=\inf\{\langle v, \gamma (0)\rangle +\int_0^TL(t, \gamma (t), {\dot
\gamma}(t))\, dt; \gamma \in C^1([0, T), M); \gamma(T)=x\}, \] where
$M=\mathbb{R}^d$, $T>0$, and $L:M\times M \to \mathbb{R}$ is a suitable
Lagrangian. We also consider the stochastic counterpart:
\begin{align*}%\tag{$\star$}
\underline{B}_T^s(\mu,\nu):=\inf\left\{\mathbb{E}\left[\langle V,X_0\rangle
+\int_0^T L(t, X,\beta(t,X))\,dt\right]; X\in \mathcal{A}, V\sim\mu,X_T\sim
\nu\right\}
  \end{align*} where $\mathcal{A}$ is the set of stochastic processes
satisfying $dX=\beta_X(t,X)\,dt+ dW_t,$ for some drift $\beta_X(t,X)$, and
where $W_t$ is $\sigma(X_s:0\le s\le t)$-Brownian motion. While inf-convolution
allows us to easily obtain Hopf-Lax formulas on Wasserstein space for cost
minimizing transports, this is not the case for total cost maximizing
transports, which actually are sup-inf problems. However, in the case where the
Lagrangian $L$ is jointly convex on phase space, Bolza-type dualities --well
known in the deterministic case but novel in the stochastic case--transform
sup-inf problems to sup-sup settings. Hopf-Lax formulas relate optimal
ballistic transports to those associated with dynamic fixed-end transports
studied by Bernard-Buffoni and Fathi-Figalli in the deterministic case, and by
Mikami-Thieullen in the stochastic setting. We also write Eulerian formulations
and point to links with the theory of mean field games.
