In this paper, we consider self-stabilization and its weakened form called pseudo-stabilization. We study conditions under which (pseudo-and self-) stabilizing leader election is solvable in networks subject to frequent topological changes. To model such an high dynamics, we use the dynamic graph (DG) paradigm and study a taxonomy of nine important DG classes. Our results show that self-stabilizing leader election can be achieved only in the classes where all processes are sources. However, we also show that, among those classes, the convergence time of pseudo-and so self-stabilizing solutions can only be bounded in the class where all sources are actually timely. Furthermore, even pseudo-stabilizing leader election cannot be solved in all remaining classes, except in the class where at least one process is a timely source. We illustrate this latter claim by proposing a pseudo-stabilizing leader election algorithm for this class. We also show that in this class, the convergence time of pseudo-stabilizing leader election algorithms cannot be bounded. Nevertheless, we show that our solution is speculative since its convergence time can be bounded when the dynamics is not too erratic, precisely when all processes are timely sources.