  We introduce traveling Floquet valley Hall edge solitons in a genuinely
continuous system consisting of a waveguide array with a dynamically varying
domain wall between two honeycomb structures exhibiting broken inversion
symmetry. Inversion symmetry in our system is broken due to periodic and
out-of-phase longitudinal modulation of the refractive index applied to the
constituent sublattices of the honeycomb structure. By combining two honeycomb
arrays with different initial phases of refractive index modulation we create a
dynamically changing domain wall that supports localized linear Floquet edge
states despite the fact that on average two sublattices in each honeycomb
structure forming the domain wall have the same refractive index. In the
presence of focusing nonlinearity, bright or dark Floquet edge solitons may
bifurcate from such linear Floquet edge states. We numerically identified a
family of these solitons and compared them with the results obtained from the
analytical approach, which involved averaging over one longitudinal period in
the evolution coordinate. These solitons exhibit localization in both spatial
directions - along the interface due to nonlinear self-action and across the
interface as the edge states - that allows them to travel along the domain wall
over long distances without noticeable shape variations.
