Parts (1), (2), (3) are local on $S$ hence we may and do assume $S$ is affine. Since $a$ is quasi-compact we conclude that $X$ is quasi-compact. Since $\mathcal{F}^\bullet$ is locally bounded, we conclude that $\mathcal{F}^\bullet$ is bounded. \medskip\noindent For (1) and (2) we can use the first spectral sequence $R^pa_*\mathcal{F}^q \Rightarrow R^{p + q}a_*\mathcal{F}^\bullet$ of Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}. Combining Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherence-higher-direct-images} and Homology, Lemma \ref{homology-lemma-biregular-ss-converges} we conclude. \medskip\noindent Let us prove (3) by the induction principle of Cohomology of Schemes, Lemma \ref{coherent-lemma-induction-principle}. Namely, for a quasi-compact open of $U$ of $X$ consider the condition that $R(a|_U)_*(\mathcal{F}^\bullet|_U)$ has finite tor dimension. If $U, V$ are quasi-compact open in $X$, then we have a relative Mayer-Vietoris distinguished triangle $$ R(a|_{U \cup V})_*\mathcal{F}^\bullet|_{U \cup V} \to R(a|_U)_*\mathcal{F}^\bullet|_U \oplus R(a|_V)_*\mathcal{F}^\bullet|_V \to R(a|_{U \cap V})_*\mathcal{F}^\bullet|_{U \cap V} \to $$ by Cohomology, Lemma \ref{cohomology-lemma-unbounded-relative-mayer-vietoris}. By the behaviour of tor amplitude in distinguished triangles (see Cohomology, Lemma \ref{cohomology-lemma-cone-tor-amplitude}) we see that if we know the result for $U$, $V$, $U \cap V$, then the result holds for $U \cup V$. This reduces us to the case where $X$ is affine. In this case we have $$ Ra_*\mathcal{F}^\bullet = a_*\mathcal{F}^\bullet $$ by Leray's acyclicity lemma (Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity}) and the vanishing of higher direct images of quasi-coherent modules under an affine morphism (Cohomology of Schemes, Lemma \ref{coherent-lemma-relative-affine-vanishing}). Since $\mathcal{F}^n$ is $S$-flat by assumption and $X$ affine, the modules $a_*\mathcal{F}^n$ are flat for all $n$. Hence $a_*\mathcal{F}^\bullet$ is a bounded complex of flat $\mathcal{O}_S$-modules and hence has finite tor dimension. \medskip\noindent Proof of part (5). Denote $a' : (X, a^{-1}\mathcal{O}_S) \to (S, \mathcal{O}_S)$ the obvious flat morphism of ringed spaces. Part (5) says that $$ K \otimes_{\mathcal{O}_S}^\mathbf{L} Ra'_*\mathcal{F}^\bullet = Ra'_*(L(a')^*K \otimes_{a^{-1}\mathcal{O}_S}^\mathbf{L} \mathcal{F}^\bullet) $$ Thus Cohomology, Equation (\ref{cohomology-equation-projection-formula-map}) gives a functorial map from the left to the right and we want to show this map is an isomorphism. This question is local on $S$ hence we may and do assume $S$ is affine. The rest of the proof is {\it exactly} the same as the proof of Lemma \ref{lemma-cohomology-base-change} except that we have to show that the functor $K \mapsto Ra'_*(L(a')^*K \otimes_{a^{-1}\mathcal{O}_S}^\mathbf{L} \mathcal{F}^\bullet)$ commutes with direct sums. This is where we will use $\mathcal{F}^n$ has the structure of a quasi-coherent $\mathcal{O}_X$-module. Namely, observe that $K \mapsto L(a')^*K \otimes_{a^{-1}\mathcal{O}_S}^\mathbf{L} \mathcal{F}^\bullet$ commutes with arbitrary direct sums. Next, if $\mathcal{F}^\bullet$ consists of a single quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}^\bullet = \mathcal{F}^n[-n]$ then we have $L(a')^*G \otimes_{a^{-1}\mathcal{O}_S}^\mathbf{L} \mathcal{F}^\bullet = La^*K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{F}^n[-n]$, see Cohomology, Lemma \ref{cohomology-lemma-variant-derived-pullback}. Hence in this case the commutation with direct sums follows from Lemma \ref{lemma-quasi-coherence-pushforward-direct-sums}. Now, in general, since $S$ is affine (hence $X$ quasi-compact) and $\mathcal{F}^\bullet$ is locally bounded, we see that $$ \mathcal{F}^\bullet = (\mathcal{F}^a \to \ldots \to \mathcal{F}^b) $$ is bounded. Arguing by induction on $b - a$ and considering the distinguished triangle $$ \mathcal{F}^b[-b] \to (\mathcal{F}^a \to \ldots \to \mathcal{F}^b) \to (\mathcal{F}^a \to \ldots \to \mathcal{F}^{b - 1}) \to \mathcal{F}^b[-b + 1] $$ the proof of this part is finished. Some details omitted. \medskip\noindent Proof of part (4). Let $a' : (X, a^{-1}\mathcal{O}_S) \to (S, \mathcal{O}_S)$ be as above. Since $\mathcal{F}^\bullet$ is a locally bounded complex of flat $a^{-1}\mathcal{O}_S$-modules we see the complex $a^{-1}\mathcal{G} \otimes_{a^{-1}\mathcal{O}_S} \mathcal{F}^\bullet$ represents $L(a')^*\mathcal{G} \otimes_{a^{-1}\mathcal{O}_S}^\mathbf{L} \mathcal{F}^\bullet$ in $D(a^{-1}\mathcal{O}_S)$. Hence (4) follows from (5).