A series expression is developed for the space-time fourth moment of a beamed field propagating through a rigid random medium that is moving relative to the transmitted field. Useful approximations for the space-time correlation of intensity fluctuations are extracted as two special cases of the fourth moment series: a low spatial-temporal-frequency region and a high spatial-temporal-frequency region. The same statistics are also derived for the case of spatially incoherent sources. The parabolic wave equation, the Markov approximation, and Gaussian refractive index fluctuations are assumed. The results are derived using path integral methods.