This lemma is a generalization of Lemma \ref{lemma-punctual-base-change} to objects of the derived category; the assertion of our lemma is true because in Lemma \ref{lemma-punctual-base-change} the scheme $X$ over $K$ is arbitrary. We strongly urge the reader to skip the laborious proof (alternative: read only the last paragraph). \medskip\noindent We may represent $E$ by a bounded below K-flat complex $\mathcal{E}^\bullet$ consisting of flat $\mathbf{Z}/n\mathbf{Z}$-modules. See Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-bounded-below-tor-amplitude}. Choose an integer $b$ such that $H^i(F) = 0$ for $i < b$. Choose a large integer $N$ and consider the short exact sequence $$ 0 \to \sigma_{\geq N + 1}\mathcal{E}^\bullet \to \mathcal{E}^\bullet \to \sigma_{\leq N}\mathcal{E}^\bullet \to 0 $$ of stupid truncations. This produces a distinguished triangle $E'' \to E \to E' \to E''[1]$ in $D(X_\etale, \mathbf{Z}/n\mathbf{Z})$. For fixed $F$ both sides of the arrow in the statement of the lemma are exact functors in $E$. Observe that $$ p^{-1}E'' \otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L} (f')^{-1}Rg_*F \quad\text{and}\quad Rh_*(h^{-1}p^{-1}E'' \otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L} e^{-1}F) $$ are sitting in degrees $\geq N + b$. Hence, if we can prove the lemma for the object $E'$, then we see that the lemma holds in degrees $\leq N + b$ and we will conclude. Some details omitted. Thus we may assume $E$ is represented by a bounded complex of flat $\mathbf{Z}/n\mathbf{Z}$-modules. Doing another argument of the same nature, we may assume $E$ is given by a single flat $\mathbf{Z}/n\mathbf{Z}$-module $\mathcal{E}$. \medskip\noindent Next, we use the same arguments for the variable $F$ to reduce to the case where $F$ is given by a single sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules $\mathcal{F}$. Say $\mathcal{F}$ is annihilated by an integer $m | n$. If $\ell$ is a prime number dividing $m$ and $m > \ell$, then we can look at the short exact sequence $0 \to \mathcal{F}[\ell] \to \mathcal{F} \to \mathcal{F}/\mathcal{F}[\ell] \to 0$ and reduce to smaller $m$. This finally reduces us to the case where $\mathcal{F}$ is annihilated by a prime number $\ell$ dividing $n$. In this case observe that $$ p^{-1}\mathcal{E} \otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L} (f')^{-1}Rg_*\mathcal{F} = p^{-1}(\mathcal{E}/\ell \mathcal{E}) \otimes_{\mathbf{F}_\ell}^\mathbf{L} (f')^{-1}Rg_*\mathcal{F} $$ by the flatness of $\mathcal{E}$. Similarly for the other term. This reduces us to the case where we are working with sheaves of $\mathbf{F}_\ell$-vector spaces which is discussed \medskip\noindent Assume $\ell$ is a prime number invertible in $K$. Assume $\mathcal{E}$, $\mathcal{F}$ are sheaves of $\mathbf{F}_\ell$-vector spaces on $X_\etale$ and $T_\etale$. We want to show that $$ p^{-1}\mathcal{E} \otimes_{\mathbf{F}_\ell} (f')^{-1}R^qg_*\mathcal{F} \longrightarrow R^qh_*(h^{-1}p^{-1}\mathcal{E} \otimes_{\mathbf{F}_\ell} e^{-1}\mathcal{F}) $$ is an isomorphism for every $q \geq 0$. This question is local on $X$ hence we may assume $X$ is affine. We can write $\mathcal{E}$ as a filtered colimit of constructible sheaves of $\mathbf{F}_\ell$-vector spaces on $X_\etale$, see Lemma \ref{lemma-torsion-colimit-constructible}. Since tensor products commute with filtered colimits and since higher direct images do too (Lemma \ref{lemma-relative-colimit}) we may assume $\mathcal{E}$ is a constructible sheaf of $\mathbf{F}_\ell$-vector spaces on $X_\etale$. Then we can choose an integer $m$ and finite and finitely presented morphisms $\pi_i : X_i \to X$, $i = 1, \ldots, m$ such that there is an injective map $$ \mathcal{E} \to \bigoplus\nolimits_{i = 1, \ldots, m} \pi_{i, *}\mathbf{F}_\ell $$ See Lemma \ref{lemma-constructible-maps-into-constant-general}. Observe that the direct sum is a constructible sheaf as well (Lemma \ref{lemma-finite-pushforward-constructible}). Thus the cokernel is constructible too (Lemma \ref{lemma-constructible-abelian}). By dimension shifting, i.e., induction on $q$, on the category of constructible sheaves of $\mathbf{F}_\ell$-vector spaces on $X_\etale$, it suffices to prove the result for the sheaves $\pi_{i, *}\mathbf{F}_\ell$ (details omitted; hint: start with proving injectivity for $q = 0$ for all constructible $\mathcal{E}$). To prove this case we extend the diagram of the lemma to $$ \xymatrix{ X_i \ar[d]^{\pi_i} & X'_i \ar[l]^{p_i} \ar[d]^{\pi'_i} & Y_i \ar[l]^{h_i} \ar[d]^{\rho_i} \\ X \ar[d] & X' \ar[l]^p \ar[d]_{f'} & Y \ar[l]^h \ar[d]^e \\ \Spec(K) & S' \ar[l] & T \ar[l]_g } $$ with all squares cartesian. In the equations below we are going to use that $R\pi_{i, *} = \pi_{i, *}$ and similarly for $\pi'_i$, $\rho_i$, we are going to use the Leray spectral sequence, we are going to use Lemma \ref{lemma-finite-pushforward-commutes-with-base-change}, and we are going to use Lemma \ref{lemma-projection-formula-proper-mod-n} (although this lemma is almost trivial for finite morphisms) for $\pi_i$, $\pi'_i$, $\rho_i$. Doing so we see that \begin{align*} p^{-1}\pi_{i, *}\mathbf{F}_\ell \otimes_{\mathbf{F}_\ell} (f')^{-1}R^qg_*\mathcal{F} & = \pi'_{i, *}\mathbf{F}_\ell \otimes_{\mathbf{F}_\ell} (f')^{-1}R^qg_*\mathcal{F} \\ & = \pi'_{i, *}((\pi'_i)^{-1} (f')^{-1}R^qg_*\mathcal{F}) \end{align*} Similarly, we have \begin{align*} R^qh_*(h^{-1}p^{-1} \pi_{i, *}\mathbf{F}_\ell \otimes_{\mathbf{F}_\ell} e^{-1}\mathcal{F}) & = R^qh_*(\rho_{i, *}\mathbf{F}_\ell \otimes_{\mathbf{F}_\ell} e^{-1}\mathcal{F}) \\ & = R^qh_*(\rho_i^{-1}e^{-1}\mathcal{F}) \\ & = \pi'_{i, *}R^qh_{i, *} \rho_i^{-1}e^{-1}\mathcal{F}) \end{align*} Simce $R^qh_{i, *} \rho_i^{-1}e^{-1}\mathcal{F} =  (\pi'_i)^{-1} (f')^{-1}R^qg_*\mathcal{F}$ by Lemma \ref{lemma-punctual-base-change} we conclude.