This talk discusses a new non-asymptotic, local approach to quantitative propagation of chaos for a wide class of mean field diffusive dynamics. For a system of interacting particles, the relative entropy between the marginal law of particles and its limiting product measure is shown to be at each time, as long as the same is true at time zero. A simple Gaussian example shows that this rate is optimal. The main assumption is that the limiting measure obeys a certain functional inequality, which is shown to encompass many potentially irregular but not too singular finite-range interactions, as well as some infinite-range interactions. This unifies the previously disparate cases of Lipschitz versus bounded measurable interactions, improving the best prior bounds of which were deduced from global estimates involving all particles. We also cover a class of models, including all uniformly continuous interactions, for which qualitative propagation of chaos and even well-posedness of the limiting dynamics were previously unknown. At the center of a new approach is a differential inequality, derived from a form of the BBGKY hierarchy, which bounds the -particle entropy in terms of the -particle entropy.