Represent $K$ by a K-injective complex $\mathcal{J}^\bullet$ of sheaves of $\Lambda$-modules on $X'_\etale$. Choose a quasi-isomorphism $g'_!\mathcal{J}^\bullet \to \mathcal{I}^\bullet$ to a K-injective complex $\mathcal{I}^\bullet$ of sheaves of $\Lambda$-modules on $X_\etale$. Then we can consider the map $$ g_!f'_*\mathcal{J}^\bullet = g_!f'_!\mathcal{J}^\bullet = f_!g'_!\mathcal{J}^\bullet = f_*g'_!\mathcal{J}^\bullet \to f_*\mathcal{I}^\bullet $$ where the first and third equality come from Lemma \ref{lemma-proper-f-shriek} and the second equality comes from Lemma \ref{lemma-f-shriek-composition} which tells us that both $g_! \circ f'_!$ and $f_! \circ g'_!$ are equal to $(g \circ f')_! = (f \circ g')_!$ as subsheaves of $(g \circ f')_* = (f \circ g')_*$. To finish the proof it suffices to show that $f_*g'_!\mathcal{J}^\bullet \to f_*\mathcal{I}^\bullet$ is a quasi-isomorphism. \medskip\noindent The question is local on $Y$. Hence we may assume that the dimension of fibres of $f$ is bounded, see Morphisms, Lemma \ref{morphisms-lemma-morphism-finite-type-bounded-dimension}. Then we see that $Rf_*$ has finite cohomological dimension, see \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-cohomological-dimension-proper}. Hence by Derived Categories, Lemma \ref{derived-lemma-unbounded-right-derived}, if we show that $R^qf_*(g'_!\mathcal{J}) = 0$ for $q > 0$ and any injective sheaf of $\Lambda$-modules $\mathcal{J}$ on $X'_\etale$, then the result follows. \medskip\noindent The stalk of $R^qf_*(g'_!\mathcal{J})$ at a geometric point $\overline{y}$ is equal to $H^q(X_{\overline{y}}, g'_!\mathcal{J}|_{X_{\overline{y}}})$ by \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-proper-base-change-stalk}. Since formation of $g'_!$ commutes with base change this is equal to $$ H^q(X_{\overline{y}}, g'_{\overline{y}, !}(\mathcal{J}|_{X'_{\overline{y}}})) $$ Since $Y' \to Y$ is locally quasi-finite, we see that $X'_{\overline{y}}$ is a disjoint union of the fibres $X'_{\overline{y}'}$ at geometric points $\overline{y}'$ of $Y'$ lying over $\overline{y}$. Hence we get $$ H^q(X_{\overline{y}}, \bigoplus g'_{\overline{y}', !}(\mathcal{J}|_{X'_{\overline{y}'}})) $$ for example by Lemma \ref{lemma-colim-f-shriek-separated} (but it is also obvious from the definition of $g'_{\overline{y}, !}$ in Section \ref{section-compact-support}). Since taking \'etale cohomology over $X_{\overline{y}}$ commutes with direct sums (\'Etale Cohomology, Theorem \ref{etale-cohomology-theorem-colimit}) we conclude it suffices to show that $$ H^q(X_{\overline{y}}, g'_{\overline{y}', !}(\mathcal{J}|_{X'_{\overline{y}'}})) $$ is zero. Observe that $g_{\overline{y}'} : X'_{\overline{y}'} \to X_{\overline{y}}$ is a morphism between proper scheme over $\overline{y}$ and hence is proper itself. As it is locally quasi-finite as well we conclude that $g_{\overline{y}'}$ is finite. Thus we see that $g'_{\overline{y}', !} = g'_{\overline{y}', *} = Rg'_{\overline{y}', *}$. By Leray we conlude that we have to show $$ H^q(X'_{\overline{y}'}, \mathcal{J}|_{X'_{\overline{y}'}})) $$ is zero. This of course follows from proper base change (in the form we quoted it before) as the higher direct images of $\mathcal{J}$ under $f'$ are zero.