We introduce a new class of distributionally robust optimization problems under decision-dependent ambiguity sets. In particular, as our ambiguity sets we consider balls centered on a decision-dependent probability distribution. The balls are based on a class of earth mover's distances that includes both the total variation distance and the Wasserstein metrics. We discuss the main computational challenges in solving the problems of interest, and provide an overview of various settings leading to tractable formulations. Some of the arising side results are also of independent interest, including mathematical programming expressions for robustified risk measures in a discrete space. Finally, we rely on state-of-the-art modeling techniques from machine scheduling and humanitarian logistics to arrive at potentially practical applications.