The last couple of decades have witnessed tremendous experimental progress in the study of Floquet media: periodic materials whose properties are altered by time-periodic forcing. Much of the theory is based on discrete or few-band models, while the corresponding continuum PDE models remain less understood. In this talk, we will focus on the dynamical question of dispersion. Near Dirac points, the relevant effective models are non-autonomous Floquet-Dirac equations, whose long-time behavior is governed by the quasi-energy structure of the monodromy operator. We will discuss recent results showing that periodic driving can produce anomalously slow dispersive decay through highly degenerate quasi-energy bands, and some interesting algebraic structure. To conclude, We will briefly mention how such estimates are connected to the metastability of Floquet edge modes, in both PDE and discrete systems, via a Fermi Golden Rule-type mechanism.