Setup: Let $X,Y$ be quasi-compact quasi-separated schemes defined over a field $k$. If necessary, you can also assume that $X,Y$ are noetherian, but I don't want to assume that $X,Y$ have the resolution property. Let $A,A'$ be quasi-coherent sheaves on $X$ and $B,B'$ quasi-coherent sheaves on $Y$. Assume that $A,B$ are of finite presentation (i.e. coherent in the noetherian case). Recall that the external tensor product is defined by $A \boxtimes B := \mathrm{pr}_X^* A \otimes_{\mathcal{O}_{X \times_k Y}} \mathrm{pr}_Y^* B$.

My question: Is there a natural isomorphism

$$\bigoplus_{p+q=n} \mathrm{Ext}^p_X(A,A') \otimes_k \mathrm{Ext}^q_Y(B,B') \cong \mathrm{Ext}^n_{X \times_k Y}(A \boxtimes B,A' \boxtimes B') ~ ?$$

I can prove this for $n=0$. It is also true when $A=\mathcal{O}_X$ and $B=\mathcal{O}_Y$ (see MO/34673), hence more generally when $A$ and $B$ are locally free of finite rank. For the general case, my idea is to take injective resolutions $I^*$ of $A'$ and $J^*$ of $B'$, and hope that the total complex of the double complex $I^* \boxtimes J^*$ is an injective or at least flasque resolution of $A' \boxtimes B'$ in order to apply Künneth's Theorem for complexes. But I'm not sure if the external tensor product of flasque sheaves is flasque. They can be chosen to be quasi-coherent in the noetherian case, which probably makes life easier.

In any case I think that this must be well-known. A reference would be appreciated. Interestingly, EGA III$_2$ §6 treats "Functeurs *Tor* locaux et globaux; formule de Künneth", but in EGA III$_1$ this was announced as "Foncteurs *Tor et Ext* locaux et globaux; formule de Künneth.".