It is shown that in a moving, lossless, dispersive dielectric, the group velocity is distinct from the energy velocity defined as the ratio of the Poynting vector and the energy density. The total energy flux is equal to the product of the total energy density and the group velocity. It consists of the electromagnetic flux represented by the Poynting vector and the particle energy flux associated with the motion of particles in the medium. Only the group velocity is shown to obey Einstein's velocity addition theorem rigorously for a moving dispersive dielectric. When a dielectric which is only frequency dispersive in its own rest frame is set in motion, it is shown that the dielectric displays properties characteristic of spatial dispersion.