As in the proof of Lemma \ref{lemma-limit-essentially-finite-presentation} we may first write $R = \colim R_\lambda$ as a directed colimit of local $\mathbf{Z}$-algebras which are essentially of finite type. Denote $\mathfrak p_\lambda$ the maximal ideal of $R_\lambda$. Next, we may assume that for some $\lambda_1 \in \Lambda$ there exist $f_{j, \lambda_1} \in R_{\lambda_1}[x_1, \ldots, x_n]$ such that $$ S = \colim_{\lambda \geq \lambda_1} S_\lambda, \text{ with } S_\lambda = (R_\lambda[x_1, \ldots, x_n]/ (f_{1, \lambda}, \ldots, f_{u, \lambda}))_{\mathfrak q_\lambda} $$ For some $\lambda_2 \in \Lambda$, $\lambda_2 \geq \lambda_1$ there exist $g_{j, \lambda_2} \in R_{\lambda_2}[x_1, \ldots, x_n, y_1, \ldots, y_m]$ with images $\overline{g}_{j, \lambda_2} \in S_{\lambda_2}[y_1, \ldots, y_m]$ such that $$ S' = \colim_{\lambda \geq \lambda_2} S'_\lambda, \text{ with } S'_\lambda = (S_\lambda[y_1, \ldots, y_m]/ (\overline{g}_{1, \lambda}, \ldots, \overline{g}_{v, \lambda}))_{\overline{\mathfrak q}'_\lambda} $$ Note that this also implies that $$ S'_\lambda = (R_\lambda[x_1, \ldots, x_n, y_1, \ldots, y_m]/ (g_{1, \lambda}, \ldots, g_{v, \lambda}))_{\mathfrak q'_\lambda} $$ Choose a presentation $$ (S')^{\oplus s} \to (S')^{\oplus t} \to M \to 0 $$ of $M$ over $S'$. Let $A \in \text{Mat}(t \times s, S')$ be the matrix of the presentation. For some $\lambda_3 \in \Lambda$, $\lambda_3 \geq \lambda_2$ we can find a matrix $A_{\lambda_3} \in \text{Mat}(t \times s, S_{\lambda_3})$ which maps to $A$. For all $\lambda \geq \lambda_3$ we let $M_\lambda = \Coker((S'_\lambda)^{\oplus s} \xrightarrow{A_\lambda} (S'_\lambda)^{\oplus t})$. \medskip\noindent With these choices, we have for each $\lambda_3 \leq \lambda \leq \mu$ that $S_\lambda \otimes_{R_{\lambda}} R_\mu \to S_\mu$ is a localization, $S'_\lambda \otimes_{S_{\lambda}} S_\mu \to S'_\mu$ is a localization, and the map $M_\lambda \otimes_{S'_\lambda} S'_\mu \to M_\mu$ is an isomorphism. This also implies that $S'_\lambda \otimes_{R_{\lambda}} R_\mu \to S'_\mu$ is a localization. Thus, since $M$ is flat over $R$ we see by Lemma \ref{lemma-colimit-eventually-flat} that for all $\lambda$ big enough the module $M_\lambda$ is flat over $R_\lambda$. Moreover, note that $ \mathfrak m = \colim \mathfrak p_\lambda $, $ S/\mathfrak mS = \colim S_\lambda/\mathfrak p_\lambda S_\lambda $, $ S'/\mathfrak mS' = \colim S'_\lambda/\mathfrak p_\lambda S'_\lambda $, and $ M/\mathfrak mM = \colim M_\lambda/\mathfrak p_\lambda M_\lambda $. Also, for each $\lambda_3 \leq \lambda \leq \mu$ we see (from the properties listed above) that $$ S'_\lambda/\mathfrak p_\lambda S'_\lambda \otimes_{S_{\lambda}/\mathfrak p_\lambda S_\lambda} S_\mu/\mathfrak p_\mu S_\mu \longrightarrow S'_\mu/\mathfrak p_\mu S'_\mu $$ is a localization, and the map $$ M_\lambda / \mathfrak p_\lambda M_\lambda \otimes_{S'_\lambda/\mathfrak p_\lambda S'_\lambda} S'_\mu /\mathfrak p_\mu S'_\mu \longrightarrow M_\mu/\mathfrak p_\mu M_\mu $$ is an isomorphism. Hence the system $(S_\lambda/\mathfrak p_\lambda S_\lambda \to S'_\lambda/\mathfrak p_\lambda S'_\lambda, M_\lambda/\mathfrak p_\lambda M_\lambda)$ is a system as in Lemma \ref{lemma-limit-module-essentially-finite-presentation} as well. We may apply Lemma \ref{lemma-colimit-eventually-flat} again because $M/\mathfrak m M$ is assumed flat over $S/\mathfrak mS$ and we see that $M_\lambda/\mathfrak p_\lambda M_\lambda$ is flat over $S_\lambda/\mathfrak p_\lambda S_\lambda$ for all $\lambda$ big enough. Thus for $\lambda$ big enough the data $R_\lambda \to S_\lambda \to S'_\lambda, M_\lambda$ satisfies the hypotheses of Lemma \ref{lemma-criterion-flatness-fibre-Noetherian}. Pick such a $\lambda$. Then $S = S_\lambda \otimes_{R_\lambda} R$ is flat over $R$, and $M = M_\lambda \otimes_{S_\lambda} S$ is flat over $S$ (since the base change of a flat module is flat).