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 of 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 cross-over from a low-density to a high-density state. For $\xi>1$ the transition to congestion is discontinuous and congestion nuclei arise.