In this paper, we develop a dynamic model of a link-based pollution permit system for congested urban transportation networks, whose set of stationary points coincides with the set of solutions to the variational inequality formulation of a previously proposed static model. We then conduct stability analysis and establish both local and global stability results and, in particular, asymptotical stability, under assumptions solely underlying the user link travel cost and origin/destination pair travel disutility functions. Finally, we propose an algorithm for the discrete time approximation of the trajectories, establish convergence, and provide numerical examples. This work is the first to study the dynamics and stability of a pollution permit system for transportation networks.