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 only be achieved in the classes where all processes are sources. 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 that result by proposing a pseudo-stabilizing leader election algorithm for the latter class. We also show that in this last case, 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.