In the presence of interactions, periodically driven quantum systems generally thermalize to infinite temperature states. However, it has recently been shown that in random kick rotors with local interactions, this long-time equilibrium is significantly delayed by operating in a region of weakly fluctuating random phases, leading to the emergence of metastable thermal ensembles. rice field. Here, when the random kinetic energy is smaller than the interaction energy, this system actually exhibits a much richer dynamic phase diagram. It involves a low-energy thermal prephase characterized by light-cone diffusion of correlations in momentum space. We have developed a hydrodynamic theory for this phase and found that it agrees very well with exact numerical simulations. Finally, we investigate the full dynamical phase diagram of the system and find that the transition to full thermalization is characterized by a relatively sharp crossover.