Suppose that $(C,J)$ and $(D,K)$ are two Grothendieck (possibly $\infty$-)sites and $f:C \to D$ is a functor such that $$f^*:Psh(D) \to Psh(C)$$ sends sheaves to sheaves. Under what conditions will $$f_!:Psh(C) \to Psh(D)$$ send sheaves to sheaves? Here, $f_!$ is the unique colimit preserving functor that sends each representable $y(c)$ to $y(f(c)).$ I am not looking for degenerate cases, but more for useful criteria to check to see if it holds in certain non-trivial examples.
Note: I am not assuming either Grothendieck topology is subcanonical, but, I am interested in this case as well.