We claim that the hypotheses of the lemma are preserved under $U$-admissible blowups. Namely, suppose $b : S' \to S$ is a $U$-admissible blowup in the quasi-coherent sheaf of ideals $\mathcal{I}$. Moreover, let $S^*_i \to S_i$ be the blowup in the quasi-coherent sheaf of ideals $\mathcal{J}_i$. Then the collection of morphisms $e'_i : S'_i = S_i \times_S S' \to S'$ and $j'_i : X_i' = X_i \times_S S' \to X \times_S S'$ satisfy conditions (1), (2), (3) for the strict transform $\mathcal{F}'$ of $\mathcal{F}$ relative to the blowup $S' \to S$. First, observe that $S_i'$ is the blowup of $S_i$ in the pullback of $\mathcal{I}$, see Divisors, Lemma \ref{divisors-lemma-flat-base-change-blowing-up}. Second, consider the blowup $S_i^{\prime *} \to S_i'$ of $S_i'$ in the pullback of the ideal $\mathcal{J}_i$. By Divisors, Lemma \ref{divisors-lemma-blowing-up-two-ideals} we get a commutative diagram $$ \xymatrix{ S_i^{\prime *} \ar[r] \ar[rd] \ar[d] & S'_i \ar[d] \\ S_i^* \ar[r] & S_i } $$ and all the morphisms in the diagram above are blowups. Hence by Divisors, Lemmas \ref{divisors-lemma-strict-transform-flat} and \ref{divisors-lemma-strict-transform-composition-blowups} we see \begin{align*} & \text{ the strict transform of }(j'_i)^*\mathcal{F}'\text{ under } S_i^{\prime *} \to S_i' \\ = & \text{ the strict transform of }j_i^*\mathcal{F}\text{ under } S_i^{\prime *} \to S_i \\ = & \text{ the strict transform of }\mathcal{F}_i'\text{ under } S_i^{\prime *} \to S_i' \\ = & \text{ the pullback of }\mathcal{F}_i^*\text{ via } X_i \times_{S_i} S_i^{\prime *} \to X_i \end{align*} which is therefore flat over $S_i^{\prime *}$ (Morphisms, Lemma \ref{morphisms-lemma-base-change-module-flat}). Having said this, we see that all observations of Remark \ref{remark-successive-blowups} apply to the problem of finding a $U$-admissible blowup such that the strict transform of $\mathcal{F}$ becomes flat over the base under assumptions as in the lemma. In particular, we may assume that $S \setminus U$ is the support of an effective Cartier divisor $D \subset S$. Another application of Remark \ref{remark-successive-blowups} combined with Divisors, Lemma \ref{divisors-lemma-characterize-effective-Cartier-divisor} shows we may assume that $S = \Spec(A)$ and $D = \Spec(A/(a))$ for some nonzerodivisor $a \in A$. \medskip\noindent Pick an $i$ and $s \in S_i$. Lemma \ref{lemma-push-ideal} implies we can find an open neighbourhood $s \in W_i \subset S_i$ and a finite type quasi-coherent ideal $\mathcal{I} \subset \mathcal{O}_S$ such that $\mathcal{I} \cdot \mathcal{O}_{W_i} = \mathcal{J}_i \mathcal{J}'_i$ for some finite type quasi-coherent ideal $\mathcal{J}'_i \subset \mathcal{O}_{W_i}$ and such that $V(\mathcal{I}) \subset V(a) = S \setminus U$. Since $S_i$ is quasi-compact we can replace $S_i$ by a finite collection $W_1, \ldots, W_n$ of these opens and assume that for each $i$ there exists a quasi-coherent sheaf of ideals $\mathcal{I}_i \subset \mathcal{O}_S$ such that $\mathcal{I}_i \cdot \mathcal{O}_{S_i} = \mathcal{J}_i \mathcal{J}'_i$ for some finite type quasi-coherent ideal $\mathcal{J}'_i \subset \mathcal{O}_{S_i}$. As in the discussion of the first paragraph of the proof, consider the blowup $S'$ of $S$ in the product $\mathcal{I}_1 \ldots \mathcal{I}_n$ (this blowup is $U$-admissible by construction). The base change of $S' \to S$ to $S_i$ is the blowup in $$ \mathcal{J}_i \cdot \mathcal{J}'_i \mathcal{I}_1 \ldots \hat{\mathcal{I}_i} \ldots \mathcal{I}_n $$ which factors through the given blowup $S_i^* \to S_i$ (Divisors, Lemma \ref{divisors-lemma-blowing-up-two-ideals}). In the notation of the diagram above this means that $S_i^{\prime *} = S_i'$. Hence after replacing $S$ by $S'$ we arrive in the situation that $j_i^*\mathcal{F}$ is flat over $S_i$. Hence $j_i^*\mathcal{F}$ is flat over $S$, see Lemma \ref{lemma-etale-flat-up-down}. By Morphisms, Lemma \ref{morphisms-lemma-flat-permanence} we see that $\mathcal{F}$ is flat over $S$.