The complexity of frugal digraph homomorphisms

For an integer $t\ge 1$, a homomorphism of a digraph G to a digraph $H$ is $t$-frugal if no more than $t$ in-neighbours of any vertex of $G$ have the same image. There is a dichotomy theorem based on structural properties when $t=1$ and $H$ has a loop at each vertex, but there is unlikely to be such a theorem for general digraphs $H$. For $t\ge 2$ we describe a structural dichotomy theorem for $t$-frugal homomorphisms of general digraphs.