  The possibility of attaining chiral edge modes under periodic driving has
spurred tremendous attention, both theoretically and experimentally, especially
in light of anomalous Floquet topological phases that feature vanishing Chern
numbers unlike any static counterpart. We here consider a periodically
modulated honeycomb lattice and experimentally relevant driving protocols,
which allows us to obtain edge modes of various character in a simple model. We
calculate the phase diagram over a wide range of parameters and recover an
anomalous topological phase with quasienergy gaps harbouring edge states with
opposite chirality. Motivated by the advances in single-site control in optical
lattices, we investigate wave packet dynamics localized at the edges in
distinct Floquet topological regimes that cannot be achieved in equilibrium. We
analyse transport properties in edge modes originating from the same bands, but
with support at different quasienergies and sublattices as well as possessing
different chiralities. We find that an anomalous Floquet topological phase can
in general generate more robust chiral edge motion than a Haldane phase. Our
results demonstrate that the rich interplay of wave packet dynamics and
topological edge states can serve as a versatile tool in ultracold quantum
gases in optical lattices.
