The problem of a point electric dipole moving over a dispersive dielectric half-space is studied. The dipole is located in the free space above the dielectric and is assumed to be time harmonic in its rest frame, oriented perpendicular to the interface, and moving parallel to it. Solutions for this problem are obtained by using integral transform techniques. Expressions for the electric and magnetic fields in the free space region are formulated for the reflected radiation field for the case of an arbitrary, dispersive dielectric and for the lateral wave and surface wave fields for the case of a lossless plasma dielectric. In the rest frame of the source, it is found that all three waves exhibit the frequency of the source and that the field patterns are distorted by the relative motion. In the rest frame of the dielectric, it is found that each of the waves exhibits some form of Doppler shift (different from the primary wave) and that their field patterns are also distorted. It is also found that the criterion for existence of the lateral wave is not modified by the relative motion of the source and dielectric but that the criterion for existence of the surface waves is greatly modified by the relative motion. In addition, for velocities greater than some critical velocity, it is found that a new type of surface wave comes into existence.