Let $M$ be an object of $D^b_{\textit{Coh}}(\mathcal{O}_X)$. In each case we will use Derived Categories of Schemes, Lemma \ref{perfect-lemma-direct-image-coherent} to show that $$ \Phi_K(M) = R\text{pr}_{2, *}( L\text{pr}_1^*M \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} K) $$ is in $D^b_{\textit{Coh}}(\mathcal{O}_Y)$. The derived tensor product $L\text{pr}_1^*M \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} K$ is a pseudo-coherent object of $D(\mathcal{O}_{X \times_S Y})$ (by Cohomology, Lemma \ref{cohomology-lemma-pseudo-coherent-pullback}, Derived Categories of Schemes, Lemma \ref{perfect-lemma-identify-pseudo-coherent-noetherian}, and Cohomology, Lemma \ref{cohomology-lemma-tensor-pseudo-coherent}) whence has coherent cohomology sheaves (by Derived Categories of Schemes, Lemma \ref{perfect-lemma-identify-pseudo-coherent-noetherian} again). In each case the supports of the cohomology sheaves $H^i(L\text{pr}_1^*M \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} K)$ is proper over $Y$ as these supports are contained in the union of the supports of the $H^i(K)$. Hence in each case it suffices to prove that this tensor product is bounded below. \medskip\noindent Case (1). By Cohomology, Lemma \ref{cohomology-lemma-variant-derived-pullback} we have $$ L\text{pr}_1^*M \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} K \cong \text{pr}_1^{-1}M \otimes_{\text{pr}_1^{-1}\mathcal{O}_X}^\mathbf{L} K $$ with obvious notation. Hence the assumption on tor dimension and the fact that $M$ has only a finite number of nonzero cohomology sheaves, implies the bound we want. \medskip\noindent Case (2) follows because here the assumption implies that $K$ has finite tor dimension as an object of $D(\text{pr}_1^{-1}\mathcal{O}_X)$ hence the argument in the previous paragraph applies. \medskip\noindent In Case (3) it is also the case that $K$ has finite tor dimension as an object of $D(\text{pr}_1^{-1}\mathcal{O}_X)$. Namely, choose affine opens $U = \Spec(A)$ and $V = \Spec(B)$ of $X$ and $Y$ mapping into the affine open $W = \Spec(R)$ of $S$. Then $K|_{U \times V}$ is given by a bounded complex of finite $A \otimes_R B$-modules $M^\bullet$. Since $A$ is a regular ring of finite dimension we see that each $M^i$ has finite projective dimension as an $A$-module (Algebra, Lemma \ref{algebra-lemma-finite-gl-dim-finite-dim-regular}) and hence finite tor dimension as an $A$-module. Thus $M^\bullet$ has finite tor dimension as a complex of $A$-modules (More on Algebra, Lemma \ref{more-algebra-lemma-complex-finite-tor-dimension-modules}). Since $X \times Y$ is quasi-compact we conclude there exist $[a, b]$ such that for every point $z \in X \times Y$ the stalk $K_z$ has tor amplitude in $[a, b]$ over $\mathcal{O}_{X, \text{pr}_1(z)}$. This implies $K$ has bounded tor dimension as an object of $D(\text{pr}_1^{-1}\mathcal{O}_X)$, see Cohomology, Lemma \ref{cohomology-lemma-tor-amplitude-stalk}. We conclude as in the previous to paragraphs. \medskip\noindent Case (4). With notation as above, the ring map $R \to B$ is flat. Hence the ring map $A \to A \otimes_R B$ is flat. Hence any projective $A \otimes_R B$-module is $A$-flat. Thus any perfect complex of $A \otimes_R B$-modules has finite tor dimension as a complex of $A$-modules and we conclude as before.