We consider a setting where two noncooperative players optimally influence the evolution of an initial spatial probability in a game-theoretic hierarchical fashion (Stackelberg differential game), so that at a specific final time the distribution of the state matches a given final target measure. We provide a sufficient condition for the existence and uniqueness of an optimal transport map and prove that it can be characterized as the gradient of some convex function. An important by-product of our formulation is that it provides a means to study a class of Stackelberg differential games where the initial and final states of the underlying system are uncertain, but drawn randomly from some probability measures.