We study the decay of primordial random motions of galaxies and galaxy clusters in an expanding universe by solving a kinetic equation for the relaxation of differential energy spectra N(E, t). Systematic dissipative energy losses are included, involving gravitational drag by, and accretion of, intergalactic matter, as well as the effect of collisions with other systems. Formal and numerical solutions are described for two distinct modes of galaxy formation in a turbulent medium, corresponding to formation at a distinct epoch and to continuous formation of galaxies. We show that any primordial random motions of galaxies at the present epoch can amount to at most a few km/sec, and that collisions at early epochs can lead to the acquisition of significant amounts of primordial angular momentum.