We study the dynamical properties of a collection of models for communication processes, characterized by a single parameter xi representing the relation between information load of the nodes and its ability to deliver this information. The critical transition to congestion reported so far occurs only for xi=1. This case is well analyzed for different network topologies. We focus on the properties of the order parameter, the susceptibility, and the time correlations when approaching the critical point. For xi<1, no transition to congestion is observed but it remains a crossover from a low-density to a high-density state. For xi>1, the transition to congestion is discontinuous and congestion nuclei arise.