Hopefully this is not too off-topic of an answer, since it looks at this question from the point of evolutionary game theory (EGT), instead of AGT.
Game theory as originally formulated by von Neumann and Morgenstern was a static theory. Hence, many of the popular equilibria concepts (Nash, Correlated, etc) are inherently static. To talk about non-static equilibria, we have to introduce some sort of dynamics. AGT often does this by considering specific reasoning (algorithms) agents might use to arrive at their decisions.
An alternative approach, and one embraced by EGT, is to consider the population dynamics of a large number of agents with very simple decision making. This usually creates non-linear dynamics in the population and places EGT as part of dynamic systems. Hence, you start to see all the crazy equilibria concepts of dynamic systems such as limit cycles or chaotic attractors pop-up as equilibrium concepts. These non-stationary equilibria are well studied in EGT, although often the analysis is purely from dynamic systems and not algorithmic.
If you are interested in EGT, then a standard (and accessible) starting point is Hofbauer and Sigmund's 2003 survey "Evolutionary game dynamics"