Recolouring reflexive digraphs

Given digraphs G and H, the colouring graph Col(G,H) has as its vertices all homomorphism of G to H. There is an arc ϕ→ϕ^′ between two homomorphisms if they differ on exactly one vertex v, and if v has a loop we also require ϕ(v)→ϕ^′(v). The recolouring problem asks if there is a path in Col(G,H) between given homomorphisms ϕ and ψ. We examine this problem in the case where G is a digraph and H is a reflexive, digraph cycle. We show that for a reflexive digraph cycle H and a reflexive digraph G, the problem of determining whether there is a path between two maps in Col(G,H) can be solved in time polynomial in G. When G is not reflexive, we show the same except for certain digraph 4-cycles H.