(This is 20.7.B in Ravi Vakil's notes)
Suppose $f:Y \to Z$ is any morphism, and $\pi: X\to Y$ is quasicompact and quasiseparated. Suppose $\mathcal{F}$ is a quasicoherent sheaf on $X$. Let
$\require{AMScd}\begin{CD} W @>{f'}>> X \\ @V{\pi'}VV @VV{\pi}V \\ Z @>>{f}> Y \end{CD}$
be a fiber diagram. Describe a natural morphism $f^*(R^i\pi_*\mathscr{F}) \to R^i\pi'_*(f')^* \mathscr{F}$ of sheaves on $Z$.
I'm not sure how to work with higher direct image sheaves at all, so help would be appreciated.