Proof.
Proof of (1). Let $\mathcal{F}$ be a sheaf on $T_{\acute{e}tale}$. Let $g : S' \to S$ be an object of $(\mathit{Sch}/S)_{\acute{e}tale}$. Consider the fibre product
\[ \xymatrix{ T' \ar[r]_{f'} \ar[d]_{g'} & S' \ar[d]^ g \\ T \ar[r]^ f & S } \]
Then we have
\[ (f_{big, *}\pi _ T^{-1}\mathcal{F})(S') = (\pi _ T^{-1}\mathcal{F})(T') = ((g'_{small})^{-1}\mathcal{F})(T') = (f'_{small, *}(g'_{small})^{-1}\mathcal{F})(S') \]
the second equality by Lemma 59.39.2. On the other hand
\[ (\pi _ S^{-1}f_{small, *}\mathcal{F})(S') = (g_{small}^{-1}f_{small, *}\mathcal{F})(S') \]
again by Lemma 59.39.2. Hence by proper base change for sheaves of sets (Lemma 59.91.5) we conclude the two sets are canonically isomorphic. The isomorphism is compatible with restriction mappings and defines an isomorphism $\pi _ S^{-1}f_{small, *}\mathcal{F} = f_{big, *}\pi _ T^{-1}\mathcal{F}$. Thus an isomorphism of functors $\pi _ S^{-1} \circ f_{small, *} = f_{big, *} \circ \pi _ T^{-1}$.
Proof of (2). There is a canonical base change map $\pi _ S^{-1}Rf_{small, *}K \to Rf_{big, *}\pi _ T^{-1}K$ for any $K$ in $D(T_{\acute{e}tale})$, see Cohomology on Sites, Remark 21.19.3. To prove it is an isomorphism, it suffices to prove the pull back of the base change map by $i_ g : \mathop{\mathit{Sh}}\nolimits (S'_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{\acute{e}tale})$ is an isomorphism for any object $g : S' \to S$ of $(\mathit{Sch}/S)_{\acute{e}tale}$. Let $T', g', f'$ be as in the previous paragraph. The pullback of the base change map is
\begin{align*} g_{small}^{-1}Rf_{small, *}K & = i_ g^{-1}\pi _ S^{-1}Rf_{small, *}K \\ & \to i_ g^{-1}Rf_{big, *}\pi _ T^{-1}K \\ & = Rf'_{small, *}(i_{g'}^{-1}\pi _ T^{-1}K) \\ & = Rf'_{small, *}((g'_{small})^{-1}K) \end{align*}
where we have used $\pi _ S \circ i_ g = g_{small}$, $\pi _ T \circ i_{g'} = g'_{small}$, and Lemma 59.99.2. This map is an isomorphism by the proper base change theorem (Lemma 59.91.12) provided $K$ is bounded below and the cohomology sheaves of $K$ are torsion.
The proof of part (3) is the same as the proof of part (2), except we use Lemma 59.92.3 instead of Lemma 59.91.12.
Proof of (4). If $f$ is finite, then the functors $f_{small, *}$ and $f_{big, *}$ are exact. This follows from Proposition 59.55.2 for $f_{small}$. Since any base change $f'$ of $f$ is finite too, we conclude from Lemma 59.99.2 part (3) that $f_{big, *}$ is exact too (as the higher derived functors are zero). Thus this case follows from part (1).
$\square$
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