The function that maps a family of probability measures to the solution of the dual entropic optimal transport problem is known as the Schrödinger map. We prove that when the cost function is $C^{k+1}$ with k in N* then this map is Lipschitz continuous from the $L^2$-Wasserstein space to the space of $C^k$ functions. Our result holds on compact domains and covers the multi-marginal case. As applications, we prove displacement smoothness of the entropic optimal transport cost and the well-posedness of certain Wasserstein gradient flows involving this functional, including the Sinkhorn divergence and a multi-species system.