  We study the low-frequency dynamics of periodically driven one-dimensional
systems hosting Floquet topological phases. We show, both analytically and
numerically, in the low-frequency limit $\Omega\to0$, the topological
invariants of a chirally-symmetric driven system exhibit universal
fluctuations. While the topological invariants in this limit nearly vanish on
average over a small range of frequencies, we find that they follow a universal
Gaussian distribution with a width that scales as $1/\sqrt{\Omega}$. We explain
this scaling based on a diffusive structure of the winding numbers of the
Floquet-Bloch evolution operator at low frequency. We also find that the
maximum quasienergy gap remains finite and scales as $\Omega^2$. Thus, we argue
that the adiabatic limit of a Floquet topological insulator is highly
structured, with universal fluctuations persisting down to very low
frequencies.
