Let $f:X\rightarrow X'$ and $g:Y\rightarrow Y'$ be morphisms of varieties. Let $\mathcal F$ be a coherent sheaf on $X$ and $\mathcal G$ be a coherent sheaf on $Y$.
Is it true that
$$(f\times g)_* (\mathcal F \boxtimes \mathcal G)=
(f_* \mathcal F) \boxtimes (g_* \mathcal G)$$?
Here I used the notation
$$\mathcal F \boxtimes \mathcal G:= \pi_X^* \mathcal F \otimes \pi_Y^* \mathcal G $$
where the $\pi$'s stand for the projections $X \leftarrow X\times Y \rightarrow Y$.
Bonus points are awarded for an answer in the relative situation, where $f,g$ are maps of varieties flat over a base variety $B$ and $\boxtimes_B$ is defined using the projections $X \leftarrow X\times_B Y \rightarrow Y$
$$(f\times g)_* (\mathcal F \boxtimes_B \mathcal G)=
(f_* \mathcal F) \boxtimes_B (g_* \mathcal G)$$
Best Answer
Let $S$ be a base scheme and $f : X \to X'$ and $g : Y \to Y'$ morphisms of $S$-schemes. If $F,G \in \mathsf{Qcoh}(X),\mathsf{Qcoh}(Y)$, we have the external tensor product $F \boxtimes G \in \mathsf{Qcoh}(X \times_S Y)$. We have a canonical homomorphism (induced by the usual adjunctions) $$f_* F \boxtimes g_* G \to (f \times g)_* (F \boxtimes G)$$ in $\mathsf{Qcoh}(X' \times_S Y')$. We ask when it is an isomorphism. Obviously, we may assume that $S$ is affine. Also, we may assume that $X'$ and $Y'$ are affine. Let us write $S=\mathrm{Spec}(R)$, $X'=\mathrm{Spec}(A)$, $Y'=\mathrm{Spec}(B)$. Then $A,B$ are $R$-algebras and $X$ (resp. $Y$) is an $A$-(resp. $B$-) scheme, and the homomorphism above becomes $$ \Gamma(X,F) \otimes_R \Gamma(Y,G) \to \Gamma(X \times_R Y,F \boxtimes G) ~~~~~ (\star)$$ in $\mathsf{Mod}(A \otimes_R B)$. The actions of $A,B$ don't matter anymore. Clearly $(\star)$ is an isomorphism when $X$ and $Y$ are affine. If $X$ is affine and $Y$ is separated and quasi-compact, then we set up the usual commutative diagram with exact columns to reduce to the affine case, provided that $\Gamma(X,F)$ is flat over $R$. As usual, the same proof can be used for generalizing to the case that $Y$ is quasi-compact and quasi-separated, and then also to the case that $X$ is quasi-compact and quasi-separated, provided that $\Gamma(Y,G)$ is flat.
Summary: When $f: X \to X'$ and $g : Y \to Y'$ are quasi-separated quasi-compact morphisms over some base $S$ and $F \in \mathsf{Qcoh}(X)$ is flat over $S$ and $G \in \mathsf{Qcoh}(Y)$ is flat over $S$, then $$f_* F \boxtimes g_* G \cong (f \times g)_* (F \boxtimes G).$$
PS: I really would like to have a reference in the literature for this fact!