The response to perturbations of a simplified transport model is studied. The model is used as a paradigmatic example of transport controlled by a critical gradient. Long-time system behaviour is diffusive when most of the system is sub-critical. However, when the critical mechanism becomes significant, it is observed to dominate the system behaviour at both short- and long-time scales. While the pulse amplitude decays in an approximately diffusive manner at long times for weakly critical situations, the pulse shape is not self-similar.