Let $f: R \to S$ be a homomorphism of commutative rings. If helpful, one can assume that $R$ is a subring of $S$. Now, let $A$ be an $S$-algebra and $M$ an $A$-bimodule. The Hochschild cohomology relative to $S$ is defined as the Ext-groups
$$ \mathrm{HH}_S^n(A,M) := \mathrm{Ext}^n_{A \otimes_S A^{op}}(A,M). $$
It is an exercise that $\mathrm{HH}^0_S(A,M)$ can be identified with $\{m \in M|am = ma , a \in A\}$, i.e., the "center" of the $A$-bimodule $M$. Although this group does not depend on whether we view $A$ as an $S$-algebra or as an $R$-algebra via $f$, the higher cohomology groups seem to behave differently in general. For example, $\mathrm{HH}^1(A,M)$ is described in terms of linear derivations, which a priori depend on the coefficient ring.
Since there is a canonical map $A \otimes_R A^{op} \to A \otimes_S A^{op}$, we obtain an induced map on derived Hom complexes:
$$ \mathrm{RHom}_{A \otimes_S A^{op}}(A,M) \to \mathrm{RHom}_{A \otimes_R A^{op}}(A,M). $$
This leads to a long exact sequence comparing the two cohomology theories. My question is: Is there a nice description of the differences between these cohomology groups? Are there any references discussing their general properties?
