Reconfiguration of homomorphisms to reflexive digraph cycles

Let G be a digraph and B be a fixed reflexive digraph cycle. Given two homomorphisms ϕ,ϕ^′:G→B, a walk from ϕ to ϕ^′ in the Hom-graph Hom(G,B) corresponds to what is often called a reconfiguration sequence of the homomorphisms. Except in the case that B contains a 4-cycle, containing an oriented cycle of algebraic girth 0, we give a polynomial time algorithm that either finds a path between two given homomorphisms or discovers an obstruction certifying the non-existence of such a path.