In the presence of interactions, periodically driven quantum systems generically thermalize to an infinite-temperature state. Recently, however, it was shown that in random kicked rotors with local interactions, this long-time equilibrium could be strongly delayed by operating in a regime of weakly fluctuating random phases, leading to the emergence of a metastable thermal ensemble. Here we show that when the random kinetic energy is smaller than the interaction energy, this system in fact exhibits a much richer dynamical phase diagram, which includes a low-energy prethermal phase characterized by a light-cone spreading of correlations in momentum space. We develop a hydrodynamic theory of this phase and find a very good agreement with exact numerical simulations. We finally explore the full dynamical phase diagram of the system and find that the transition toward full thermalization is characterized by relatively sharp crossovers.