We analyze a Wardrop equilibrium model for passenger assignment in general transit networks, including the effects of congestion over the passengers' choices. The model is based on the common-line paradigm, which is applied to general networks using a dynamic programming approach. Congestion is treated by means of a simplified bulk queue model described in the appendix. We provide a complete characterization of the set of equilibria in the common-line setting, including the conditions for existence and uniqueness. This characterization reveals the existence of ranges of flow in which a Braess-like paradox appears, and in which a flow increase does not affect the system performance as measured by transit times. The congested common-line model is used to state an equilibrium model for general transit networks, and to establish the existence of a network equilibrium.