Since the problem is local on $S$ we may assume that $S$ is a Noetherian scheme. Since a proper morphism is of finite type we see that in this case $X$ is a Noetherian scheme also. Consider the property $\mathcal{P}$ of coherent sheaves on $X$ defined by the rule $$ \mathcal{P}(\mathcal{F}) \Leftrightarrow R^pf_*\mathcal{F}\text{ is coherent for all }p \geq 0 $$ We are going to use the result of Lemma \ref{lemma-property} to prove that $\mathcal{P}$ holds for every coherent sheaf on $X$. \medskip\noindent Let $$ 0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0 $$ be a short exact sequence of coherent sheaves on $X$. Consider the long exact sequence of higher direct images $$ R^{p - 1}f_*\mathcal{F}_3 \to R^pf_*\mathcal{F}_1 \to R^pf_*\mathcal{F}_2 \to R^pf_*\mathcal{F}_3 \to R^{p + 1}f_*\mathcal{F}_1 $$ Then it is clear that if 2-out-of-3 of the sheaves $\mathcal{F}_i$ have property $\mathcal{P}$, then the higher direct images of the third are sandwiched in this exact complex between two coherent sheaves. Hence these higher direct images are also coherent by Lemma \ref{lemma-coherent-abelian-Noetherian} and \ref{lemma-coherent-Noetherian-quasi-coherent-sub-quotient}. Hence property $\mathcal{P}$ holds for the third as well. \medskip\noindent Let $Z \subset X$ be an integral closed subscheme. We have to find a coherent sheaf $\mathcal{F}$ on $X$ whose support is contained in $Z$, whose stalk at the generic point $\xi$ of $Z$ is a $1$-dimensional vector space over $\kappa(\xi)$ such that $\mathcal{P}$ holds for $\mathcal{F}$. Denote $g = f|_Z : Z \to S$ the restriction of $f$. Suppose we can find a coherent sheaf $\mathcal{G}$ on $Z$ such that (a) $\mathcal{G}_\xi$ is a $1$-dimensional vector space over $\kappa(\xi)$, (b) $R^pg_*\mathcal{G} = 0$ for $p > 0$, and (c) $g_*\mathcal{G}$ is coherent. Then we can consider $\mathcal{F} = (Z \to X)_*\mathcal{G}$. As $Z \to X$ is a closed immersion we see that $(Z \to X)_*\mathcal{G}$ is coherent on $X$ and $R^p(Z \to X)_*\mathcal{G} = 0$ for $p > 0$ (Lemma \ref{lemma-finite-pushforward-coherent}). Hence by the relative Leray spectral sequence (Cohomology, Lemma \ref{cohomology-lemma-relative-Leray}) we will have $R^pf_*\mathcal{F} = R^pg_*\mathcal{G} = 0$ for $p > 0$ and $f_*\mathcal{F} = g_*\mathcal{G}$ is coherent. Finally $\mathcal{F}_\xi = ((Z \to X)_*\mathcal{G})_\xi = \mathcal{G}_\xi$ which verifies the condition on the stalk at $\xi$. Hence everything depends on finding a coherent sheaf $\mathcal{G}$ on $Z$ which has properties (a), (b), and (c). \medskip\noindent We can apply Chow's Lemma \ref{lemma-chow-Noetherian} to the morphism $Z \to S$. Thus we get a diagram $$ \xymatrix{ Z \ar[rd]_g & Z' \ar[d]^-{g'} \ar[l]^\pi \ar[r]_i & \mathbf{P}^n_S \ar[dl] \\ & S & } $$ as in the statement of Chow's lemma. Also, let $U \subset Z$ be the dense open subscheme such that $\pi^{-1}(U) \to U$ is an isomorphism. By the discussion in Remark \ref{remark-chow-Noetherian} we see that $i' = (i, \pi) : Z' \to \mathbf{P}^n_Z$ is a closed immersion. Hence $$ \mathcal{L} = i^*\mathcal{O}_{\mathbf{P}^n_S}(1) \cong (i')^*\mathcal{O}_{\mathbf{P}^n_Z}(1) $$ is $g'$-relatively ample and $\pi$-relatively ample (for example by Morphisms, Lemma \ref{morphisms-lemma-characterize-ample-on-finite-type}). Hence by Lemma \ref{lemma-kill-by-twisting} there exists an $n \geq 0$ such that both $R^p\pi_*\mathcal{L}^{\otimes n} = 0$ for all $p > 0$ and $R^p(g')_*\mathcal{L}^{\otimes n} = 0$ for all $p > 0$. Set $\mathcal{G} = \pi_*\mathcal{L}^{\otimes n}$. Property (a) holds because $\pi_*\mathcal{L}^{\otimes}|_U$ is an invertible sheaf (as $\pi^{-1}(U) \to U$ is an isomorphism). Properties (b) and (c) hold because by the relative Leray spectral sequence (Cohomology, Lemma \ref{cohomology-lemma-relative-Leray}) we have $$ E_2^{p, q} = R^pg_* R^q\pi_*\mathcal{L}^{\otimes n} \Rightarrow R^{p + q}(g')_*\mathcal{L}^{\otimes n} $$ and by choice of $n$ the only nonzero terms in $E_2^{p, q}$ are those with $q = 0$ and the only nonzero terms of $R^{p + q}(g')_*\mathcal{L}^{\otimes n}$ are those with $p = q = 0$. This implies that $R^pg_*\mathcal{G} = 0$ for $p > 0$ and that $g_*\mathcal{G} = (g')_*\mathcal{L}^{\otimes n}$. Finally, applying the previous Lemma \ref{lemma-locally-projective-pushforward} we see that $g_*\mathcal{G} = (g')_*\mathcal{L}^{\otimes n}$ is coherent as desired.