The temporal coherence of the scattered light is analyzed in the case of a point aperture, as well as in the Fraunhofer region of an aperture which is placed in front of a uniformly moving nonabsorbing random phase screen illuminated by a quasimonochromatic plane wave. It is shown that the temporal coherence of the scattered light is affected by not only the motion of the phase screen but also the statistical properties of the random phase screen. Moreover, the coherence of the scattered light in the frequency domain is considered in the case of a polychromatic plane wave. Theoretical considerations are made with a particular emphasis on the evaluation of the spectral changes of the scattered light occurring on its free space propagation from the phase screen to its far zone. It is shown that the spectral changes affected by both the motion of the random phase screen and its statistical properties are present. Both strongly and weakly scattering phase screens are considered. Analytic or closed-form expressions for both the temporal coherence and the spectrum of the scattered light are obtained, and numerical examples are presented.