Consider a commutative diagram

$\require{AMScd}$
\begin{CD}
X' @>{f'}>> X\\
@V{p'} VV @VV{p} V\\
S' @>{f}>> S
\end{CD}

of schemes (or even locally ringed spaces). If $\mathcal{F}$ is any $\mathcal{O}_X$-module, then there is a base change morphism
$$ f^* R^i p_* \mathcal F \to R^i(p')_* (f')^* \mathcal F.   $$
The flat base change theorem says that if all objects are schemes and if $\mathcal F$ is quasicoherent, then base change is an isomorphism. Is there a nice counterexample if $\mathcal F$ is *not* quasicoherent?