Choose a commutative diagram $$ \xymatrix{ X \ar[d]_f & X' \ar[l] \ar[d]_{f'} & Y \ar[l]^h \ar[d]^e \\ S & S' \ar[l] & T \ar[l]_g } $$ with $X' = X \times_S S'$ and $Y = X' \times_{S'} T$ and $g$ quasi-compact and quasi-separated, and an abelian sheaf $\mathcal{F}$ on $T_\etale$ annihilated by $n$ such that the base change map $(f')^{-1}R^qg_*\mathcal{F} \to R^qh_*e^{-1}\mathcal{F}$ is not an isomorphism. Of course we may and do replace $S'$ by an affine open of $S'$; this implies that $T$ is quasi-compact and quasi-separated. By Lemma \ref{lemma-base-change-q-injective} we see $(f')^{-1}R^qg_*\mathcal{F} \to R^qh_*e^{-1}\mathcal{F}$ is injective. Pick a geometric point $\overline{x}$ of $X'$ and an element $\xi$ of $(R^qh_*q^{-1}\mathcal{F})_{\overline{x}}$ which is not in the image of the map $((f')^{-1}R^qg_*\mathcal{F})_{\overline{x}} \to (R^qh_*e^{-1}\mathcal{F})_{\overline{x}}$. \medskip\noindent Consider a morphism $\pi : T' \to T$ with $T'$ quasi-compact and quasi-separated and denote $\mathcal{F}' = \pi^{-1}\mathcal{F}$. Denote $\pi' : Y' = Y \times_T T' \to Y$ the base change of $\pi$ and $e' : Y' \to T'$ the base change of $e$. Picture $$ \vcenter{ \xymatrix{ X' \ar[d]_{f'} & Y \ar[l]^h \ar[d]^e & Y' \ar[l]^{\pi'} \ar[d]^{e'} \\ S' & T \ar[l]_g & T' \ar[l]_\pi } } \quad\text{and}\quad \vcenter{ \xymatrix{ X' \ar[d]_{f'} & & Y' \ar[ll]^{h' = h \circ \pi'} \ar[d]^{e'} \\ S' & & T' \ar[ll]_{g' = g \circ \pi} } } $$ Using pullback maps we obtain a canonical commutative diagram $$ \xymatrix{ (f')^{-1}R^qg_*\mathcal{F} \ar[r] \ar[d] & (f')^{-1}R^qg'_*\mathcal{F}' \ar[d] \\ R^qh_*e^{-1}\mathcal{F} \ar[r] & R^qh'_*(e')^{-1}\mathcal{F}' } $$ of abelian sheaves on $X'$. Let $P(T')$ be the property \begin{itemize} \item The image $\xi'$ of $\xi$ in $(Rh'_*(e')^{-1}\mathcal{F}')_{\overline{x}}$ is not in the image of the map $(f^{-1}R^qg'_*\mathcal{F}')_{\overline{x}} \to (R^qh'_*(e')^{-1}\mathcal{F}')_{\overline{x}}$. \end{itemize} We claim that hypotheses (1), (2), and (3) of Lemma \ref{lemma-formal-argument} hold for $P$ which proves our lemma. \medskip\noindent Condition (1) of Lemma \ref{lemma-formal-argument} holds for $P$ because the \'etale topology of a scheme and a thickening of the scheme is the same. See Proposition \ref{proposition-topological-invariance}. \medskip\noindent Suppose that $I$ is a directed set and that $T_i$ is an inverse system over $I$ of quasi-compact and quasi-separated schemes over $T$ with affine transition morphisms. Set $T' = \lim T_i$. Denote $\mathcal{F}'$ and $\mathcal{F}_i$ the pullback of $\mathcal{F}$ to $T'$, resp.\ $T_i$. Consider the diagrams $$ \vcenter{ \xymatrix{ X \ar[d]_{f'} & Y \ar[l]^h \ar[d]^e & Y_i \ar[l]^{\pi_i'} \ar[d]^{e_i} \\ S & T \ar[l]_g & T_i \ar[l]_{\pi_i} } } \quad\text{and}\quad \vcenter{ \xymatrix{ X \ar[d]_{f'} & & Y_i \ar[ll]^{h_i = h \circ \pi_i'} \ar[d]^{e_i} \\ S & & T_i \ar[ll]_{g_i = g \circ \pi_i} } } $$ as in the previous paragraph. It is clear that $\mathcal{F}'$ on $T'$ is the colimit of the pullbacks of $\mathcal{F}_i$ to $T'$ and that $(e')^{-1}\mathcal{F}'$ is the colimit of the pullbacks of $e_i^{-1}\mathcal{F}_i$ to $Y'$. By Lemma \ref{lemma-relative-colimit-general} we have $$ R^qh'_*(e')^{-1}\mathcal{F}' = \colim R^qh_{i, *}e_i^{-1}\mathcal{F}_i \quad\text{and}\quad (f')^{-1}R^qg'_*\mathcal{F}' = \colim (f')^{-1}R^qg_{i, *}\mathcal{F}_i $$ It follows that if $P(T_i)$ is true for all $i$, then $P(T')$ holds. Thus condition (2) of Lemma \ref{lemma-formal-argument} holds for $P$. \medskip\noindent The most interesting is condition (3) of Lemma \ref{lemma-formal-argument}. Assume $T'$ is a quasi-compact and quasi-separated scheme over $T$ such that $P(T')$ is true. Let $Z \subset T'$ be a closed subscheme with complement $V \subset T'$ quasi-compact. Consider the diagram $$ \xymatrix{ Y' \times_{T'} Z \ar[d]_{e_Z} \ar[r]_{i'} & Y' \ar[d]_{e'} & Y' \times_{T'} V \ar[l]^{j'} \ar[d]^{e_V} \\ Z \ar[r]^i & T' & V \ar[l]_j } $$ Choose an injective map $j^{-1}\mathcal{F}' \to \mathcal{J}$ where $\mathcal{J}$ is an injective sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules on $V$. Looking at stalks we see that the map $$ \mathcal{F}' \to \mathcal{G} = j_*\mathcal{J} \oplus i_*i^{-1}\mathcal{F}' $$ is injective. Thus $\xi'$ maps to a nonzero element of \begin{align*} & \Coker\left( ((f')^{-1}R^qg'_*\mathcal{G})_{\overline{x}} \to (R^qh'_*(e')^{-1}\mathcal{G})_{\overline{x}} \right) = \\ & \Coker\left( ((f')^{-1}R^qg'_*j_*\mathcal{J})_{\overline{x}} \to (R^qh'_*(e')^{-1}j_*\mathcal{J})_{\overline{x}} \right) \oplus \\ & \Coker\left( ((f')^{-1}R^qg'_*i_*i^{-1}\mathcal{F}')_{\overline{x}} \to (R^qh'_*(e')^{-1}i_*i^{-1}\mathcal{F}')_{\overline{x}} \right) \end{align*} by part (2) of Lemma \ref{lemma-base-change-q-injective}. If $\xi'$ does not map to zero in the second summand, then we use $$ (f')^{-1}R^qg'_*i_*i^{-1}\mathcal{F}' = (f')^{-1}R^q(g' \circ i)_*i^{-1}\mathcal{F}' $$ (because $Ri_* = i_*$ by Proposition \ref{proposition-finite-higher-direct-image-zero}) and $$ R^qh'_*(e')^{-1}i_*i^{-1}\mathcal{F} = R^qh'_*i'_*e_Z^{-1}i^{-1}\mathcal{F} = R^q(h' \circ i')_*e_Z^{-1}i^{-1}\mathcal{F}' $$ (first equality by Lemma \ref{lemma-finite-pushforward-commutes-with-base-change} and the second because $Ri'_* = i'_*$ by Proposition \ref{proposition-finite-higher-direct-image-zero}) to we see that we have $P(Z)$. Finally, suppose $\xi'$ does not map to zero in the first summand. We have $$ (e')^{-1}j_*\mathcal{J} = j'_*e_V^{-1}\mathcal{J} \quad\text{and}\quad R^aj'_*e_V^{-1}\mathcal{J} = 0, \quad a = 1, \ldots, q - 1 $$ by $BC(f, n, q - 1)$ applied to the diagram $$ \xymatrix{ X \ar[d]_f & Y' \ar[l] \ar[d]_{e'} & Y \ar[l]^{j'} \ar[d]^{e_V} \\ S & T' \ar[l] & V \ar[l]_j } $$ and the fact that $\mathcal{J}$ is injective. By the relative Leray spectral sequence for $h' \circ j'$ (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-relative-Leray}) we deduce that $$ R^qh'_*(e')^{-1}j_*\mathcal{J} = R^qh'_*j'_*e_V^{-1}\mathcal{J} \longrightarrow R^q(h' \circ j')_* e_V^{-1}\mathcal{J} $$ is injective. Thus $\xi$ maps to a nonzero element of $(R^q(h' \circ j')_* e_V^{-1}\mathcal{J})_{\overline{x}}$. Applying part (3) of Lemma \ref{lemma-base-change-q-injective} to the injection $j^{-1}\mathcal{F}' \to \mathcal{J}$ we conclude that $P(V)$ holds.