I’ll consider orbits of Borel probability measures on compact metric spaces, If the tangent derivative for such an orbit is itself a Borel measure, then the orbit is at least Holder continuous with respect to the Wasserstein metric. The Holder exponent seems to depend on the topology of the support of the measure, as well as its derivative. In fact, a metric derivative may not exist even if the orbit itself has a C^infty density in both space and time. I’ll represent some sharp results regarding the Holder exponent and the Holder norm for such orbits.