We study the problem of minimizing a smooth convex functional of a probability measure. This formulation can be used to encompass a wide range of problems and algorithms of interest in diverse areas such as reinforcement learning, variational inference, deconvolution and adversarial training. We introduce and study a class of Frank-Wolfe algorithms for solving this problem together with associated convergence guarantees which match finite dimensional optimization results. We illustrate our results in the context of Wasserstein barycenter relaxations with unconstrained support, optimal deconvolution, among others.