  Many studies have been conducted on flows of probability measures, often in
terms of gradient flows. We utilize a generalized notion of derivatives with
respect to time to model the instantaneous evolution of empirically observed
one-dimensional distributions that vary over time and develop consistent
estimates for these derivatives. Employing local Fr\'echet regression and
working in local tangent spaces with regard to the Wasserstein metric, we
derive the rate of convergence of the proposed estimators. The resulting time
dynamics are illustrated with time-varying distribution data that include
yearly income distributions and the evolution of mortality over calendar years.
