We investigate causality constraints on the time evolution of entanglement entropy after a global quench in relativistic theories. We  provide a general proof that the so-called tsunami velocity is bounded by the speed of light. We discuss various models of entanglement propagation. In particular, for free streaming models we prove an upper bound on the the tsunami velocity which is more stringent the speed of light, and smaller than what one gets for quenches in holographic theories. This highlights the importance of interactions in the spread of entanglement in many-body systems. We propose an interacting model which we believe provides an upper bound on the spread of entanglement for interacting relativistic theories.