  In this article, we provide sufficient conditions under which the controlled
vector fields solution of optimal control problems formulated on continuity
equations are Lipschitz regular in space. Our approach involves a novel
combination of mean-field approximations for infinite-dimensional multi-agent
optimal control problems, along with a careful extension of an existence result
of locally optimal Lipschitz feedbacks. The latter is based on the
reformulation of a coercivity estimate in the language of Wasserstein calculus,
which is used to obtain uniform Lipschitz bounds along sequences of
approximations by empirical measures.
