Let $f:R\to S$ be a flat homomorphism of commutative Noetherian rings. "Flat Base Change Theorem", compares the local cohomology modules $H^i_a(M) \otimes_R S$ and $H^i_{aS} (M\otimes_R S)$ for $i ∈ N_0$ and an arbitrary $R$-module $M$.
I have two questions:
Question.1.
Can one generalize this? such as: for a (flat) $S$-module instead of $S$; ($H^i_{aS} (M\otimes_R N)$)?
Question.2. (which is more important for me)

Is there a known fact about $H^i_{aS} (Hom_S (M,N)$?
In this case any (non-trivial) condition can be pose on modules and rings; (for example $R=S$)