Associated primes and finite base change

Question

Let $R$ be an integrally closed commutative Noetherian integral domain. Let $R \subseteq S$ be a ring extension such that $S$ is also an integral domain and is finite as an $R$-module. Let $I$ be an ideal of $R$. What can we say about the following two sets $$\operatorname{Ass}_R(R/I) \quad \text{ and }\quad \pi(\operatorname{Ass}_S(S/IS))$$ where $\pi: \operatorname{Spec} S \rightarrow \operatorname{Spec} R$ is the usual map of prime spectra? Is there a containment in either direction? I know they are equal if $S$ is faithfully flat over $R$, but I don't know what happens in the module-finite case.