As a first step we reduce this lemma to the case where $R$ is of finite type over $\mathbf{Z}$. By Lemma \ref{lemma-descend-faithfully-flat-finite-presentation} there exists a diagram $$ \xymatrix{ S_0 \ar[r] & S \\ R_0 \ar[u] \ar[r] & R \ar[u] } $$ where $R_0$ is of finite type over $\mathbf{Z}$, and $S_0$ is faithfully flat of finite presentation over $R_0$ such that $S = R \otimes_{R_0} S_0$. If we prove the lemma for the ring map $R_0 \to S_0$, then the lemma follows for $R \to S$ by base change, as the base change of a quasi-finite ring map is quasi-finite, see Lemma \ref{lemma-quasi-finite-base-change}. (Of course we also use that base changes of flat maps are flat and base changes of maps of finite presentation are of finite presentation.) \medskip\noindent Assume $R \to S$ is a faithfully flat ring map of finite presentation and that $R$ is Noetherian (which we may assume by the preceding paragraph). Let $W \subset \Spec(S)$ be the open set of Lemma \ref{lemma-finite-presentation-flat-CM-locus-open}. As $R \to S$ is faithfully flat the map $\Spec(S) \to \Spec(R)$ is surjective, see Lemma \ref{lemma-ff-rings}. By Lemma \ref{lemma-generic-CM-flat-finite-presentation} the map $W \to \Spec(R)$ is also surjective. Hence by replacing $S$ with a product $S_{g_1} \times \ldots \times S_{g_m}$ we may assume $W = \Spec(S)$; here we use that $\Spec(R)$ is quasi-compact (Lemma \ref{lemma-quasi-compact}), and that the map $\Spec(S) \to \Spec(R)$ is open (Proposition \ref{proposition-fppf-open}). Suppose that $\mathfrak p \subset R$ is a prime. Choose a prime $\mathfrak q \subset S$ lying over $\mathfrak p$ which corresponds to a maximal ideal of the fibre ring $S \otimes_R \kappa(\mathfrak p)$. The Noetherian local ring $\overline{S}_{\mathfrak q} = S_{\mathfrak q}/\mathfrak pS_{\mathfrak q}$ is Cohen-Macaulay, say of dimension $d$. We may choose $f_1, \ldots, f_d$ in the maximal ideal of $S_{\mathfrak q}$ which map to a regular sequence in $\overline{S}_{\mathfrak q}$. Choose a common denominator $g \in S$, $g \not \in \mathfrak q$ of $f_1, \ldots, f_d$, and consider the $R$-algebra $$ S' = S_g/(f_1, \ldots, f_d). $$ By construction there is a prime ideal $\mathfrak q' \subset S'$ lying over $\mathfrak p$ and corresponding to $\mathfrak q$ (via $S_g \to S'_g$). Also by construction the ring map $R \to S'$ is quasi-finite at $\mathfrak q$ as the local ring $$ S'_{\mathfrak q'}/\mathfrak pS'_{\mathfrak q'} = S_{\mathfrak q}/(f_1, \ldots, f_d) + \mathfrak pS_{\mathfrak q} = \overline{S}_{\mathfrak q}/(\overline{f}_1, \ldots, \overline{f}_d) $$ has dimension zero, see Lemma \ref{lemma-isolated-point-fibre}. Also by construction $R \to S'$ is of finite presentation. Finally, by Lemma \ref{lemma-grothendieck-regular-sequence} the local ring map $R_{\mathfrak p} \to S'_{\mathfrak q'}$ is flat (this is where we use that $R$ is Noetherian). Hence, by openness of flatness (Theorem \ref{theorem-openness-flatness}), and openness of quasi-finiteness (Lemma \ref{lemma-quasi-finite-open}) we may after replacing $g$ by $gg'$ for a suitable $g' \in S$, $g' \not \in \mathfrak q$ assume that $R \to S'$ is flat and quasi-finite. The image $\Spec(S') \to \Spec(R)$ is open and contains $\mathfrak p$. In other words we have shown a ring $S'$ as in the statement of the lemma exists (except possibly the faithfulness part) whose image contains any given prime. Using one more time the quasi-compactness of $\Spec(R)$ we see that a finite product of such rings does the job.