Has this form of "kind-of-dual" homomorphisms been studied?
In the case of undirected graphs, there is literature about such morphisms called graph emulators.
They are a refinement of the notion of covering graph.
A covering graph maps the neighbourhood of a vertex $u$ bijectively to the neighbourhood of $h(u)$, while an emulator is only required to do it surjectively, which seems equivalent to your question.

This notion appears in the context of characterizing graphs having planar covers or planar emulators, which yields several open problems, see for instance this paper for a survey.