  Topological theories have established a new set of rules that govern the
transport properties in a wide variety of wave-mechanical settings. In a marked
departure from the established approaches that induce Floquet topological
phases by specifically tailored discrete coupling protocols or helical lattice
motions, we introduce a new class of bimorphic Floquet topological insulators
that leverage connective chains with periodically modulated on-site potentials
to unlock new topological features in the system. In exploring a 'chain-driven'
generalization of the archetypical Floquet honeycomb lattice, we identify a
rich phase structure that can host multiple non-trivial topological phases
associated simultaneously with both Chern-type and anomalous chiral states.
Experiments carried out in photonic waveguide lattices reveal a unique and
strongly confined helical edge state that, owing to its origin in bulk flat
bands, can be set into motion in a topologically protected fashion, or halted
at will, without compromising its adherence to individual lattice sites.
