Proof of (A). Suppose $\mathcal{G}^i$ is nonzero only for $i \in [a, b]$. We may replace $X$ by a quasi-compact open neighbourhood of the union of the supports of $\mathcal{G}^i$. Hence we may assume $X$ is Noetherian. In this case $X$ and $f$ are quasi-compact and quasi-separated. Choose an approximation $P \to E$ by a perfect complex $P$ of $(X, E, -m - 1 + a)$ (possible by Theorem \ref{theorem-approximation}). Then the induced map $$ \Ext^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) \longrightarrow \Ext^i_{\mathcal{O}_X}(P, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) $$ is an isomorphism for $i \leq m$. Namely, the kernel, resp.\ cokernel of this map is a quotient, resp.\ submodule of $$ \Ext^i_{\mathcal{O}_X}(C, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) \quad\text{resp.}\quad \Ext^{i + 1}_{\mathcal{O}_X}(C, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) $$ where $C$ is the cone of $P \to E$. Since $C$ has vanishing cohomology sheaves in degrees $\geq -m - 1 + a$ these $\Ext$-groups are zero for $i \leq m + 1$ by Derived Categories, Lemma \ref{derived-lemma-negative-exts}. This reduces us to the case that $E$ is a perfect complex which is Lemma \ref{lemma-compute-ext-perfect}. The statement on boundaries is explained in the proof of Lemma \ref{lemma-compute-ext-perfect}. \medskip\noindent Proof of (B). As in the proof of (A) we may assume $X$ is Noetherian. Observe that $E$ is pseudo-coherent by Lemma \ref{lemma-identify-pseudo-coherent-noetherian}. By Lemma \ref{lemma-pseudo-coherent-hocolim} we can write $E = \text{hocolim} E_n$ with $E_n$ perfect and $E_n \to E$ inducing an isomorphism on truncations $\tau_{\geq -n}$. Let $E_n^\vee$ be the dual perfect complex (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-dual-perfect-complex}). We obtain an inverse system $\ldots \to E_3^\vee \to E_2^\vee \to E_1^\vee$ of perfect objects. This in turn gives rise to an inverse system $$ \ldots \to K_3 \to K_2 \to K_1\quad\text{with}\quad K_n = Rf_*(E_n^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{G}^\bullet) $$ perfect on $Y$, see Lemma \ref{lemma-tensor-perfect}. By Lemma \ref{lemma-compute-ext-perfect} and its proof and by the arguments in the previous paragraph (with $P = E_n$) for any quasi-coherent $\mathcal{F}$ on $Y$ we have functorial canonical maps $$ \xymatrix{ & \Ext^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) \ar[ld] \ar[rd] \\ H^i(Y, K_{n + 1} \otimes_{\mathcal{O}_Y}^\mathbf{L} \mathcal{F}) \ar[rr] & & H^i(Y, K_n \otimes_{\mathcal{O}_Y}^\mathbf{L} \mathcal{F}) } $$ which are isomorphisms for $i \leq n + a$. Let $L_n = K_n^\vee$ be the dual perfect complex. Then we see that $L_1 \to L_2 \to L_3 \to \ldots$ is a system of perfect objects in $D(\mathcal{O}_Y)$ such that for any quasi-coherent $\mathcal{F}$ on $Y$ the maps $$ \Ext^i_{\mathcal{O}_Y}(L_{n + 1}, \mathcal{F}) \longrightarrow \Ext^i_{\mathcal{O}_Y}(L_n, \mathcal{F}) $$ are isomorphisms for $i \leq n + a - 1$. This implies that $L_n \to L_{n + 1}$ induces an isomorphism on truncations $\tau_{\geq -n - a + 2}$ (hint: take cone of $L_n \to L_{n + 1}$ and look at its last nonvanishing cohomology sheaf). Thus $L = \text{hocolim} L_n$ is pseudo-coherent, see Lemma \ref{lemma-pseudo-coherent-hocolim}. The mapping property of homotopy colimits gives that $\Ext^i_{\mathcal{O}_Y}(L, \mathcal{F}) = \Ext^i_{\mathcal{O}_Y}(L_n, \mathcal{F})$ for $i \leq n + a - 3$ which finishes the proof.