Choose a surjective \'etale morphism $U \to X$ with $U$ affine. Choose a closed immersion $U \to \mathbf{A}^d_S$. Set $U_0 = S_0 \times_S U$. The complex $E_0|_{U_0}$ has tor amplitude in $[a, b]$ for some $a, b \in \mathbf{Z}$. Let $\overline{x}$ be a geometric point of $X$. We will show that the tor amplitude of $E_{\overline{x}}$ over $R$ is in $[a - d, b]$. This will finish the proof as the tor amplitude can be read off from the stalks by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-tor-amplitude-stalk} and Properties of Spaces, Theorem \ref{spaces-properties-theorem-exactness-stalks}. \medskip\noindent Let $x \in |X|$ be the point determined by $\overline{x}$. Recall that $|X| \to |S|$ is closed (by definition of proper morphisms). Since $I$ is contained in the Jacobson radical, any nonempty closed subset of $S$ contains a point of the closed subscheme $S_0$. Hence we can find a specialization $x \leadsto x_0$ in $|X|$ with $x_0 \in |X_0|$. Choose $u_0 \in U_0$ mapping to $x_0$. By Decent Spaces, Lemma \ref{decent-spaces-lemma-generalizations-lift-flat} (or by Decent Spaces, Lemma \ref{decent-spaces-lemma-specialization} which applies directly to \'etale morphisms) we find a specialization $u \leadsto u_0$ in $U$ such that $u$ maps to $x$. We may lift $\overline{x}$ to a geometric point $\overline{u}$ of $U$ lying over $u$. Then we have $E_{\overline{x}} = (E|_U)_{\overline{u}}$. \medskip\noindent Write $U = \Spec(A)$. Then $A$ is a flat, finitely presented $R$-algebra which is a quotient of a polynomial $R$-algebra in $d$-variables. The restriction $E|_U$ corresponds (by Derived Categories of Spaces, Lemmas \ref{spaces-perfect-lemma-pseudo-coherent}, \ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}, and \ref{spaces-perfect-lemma-descend-pseudo-coherent} and Derived Categories of Schemes, Lemma \ref{perfect-lemma-affine-compare-bounded} and \ref{perfect-lemma-pseudo-coherent-affine}) to a pseudo-coherent object $K$ of $D(A)$. Observe that $E_0$ corresponds to $K \otimes_A^\mathbf{L} A/IA$. Let $\mathfrak q \subset \mathfrak q_0 \subset A$ be the prime ideals corresponding to $u \leadsto u_0$. Then $$ E_{\overline{x}} = (E|_U)_{\overline{u}} = E_u \otimes_{\mathcal{O}_{U, u}}^\mathbf{L} \mathcal{O}_{U, \overline{u}} = K_{\mathfrak q} \otimes_{A_\mathfrak q}^\mathbf{L} A_{\mathfrak q}^{sh} $$ (some details omitted). Since $A_\mathfrak q \to A_\mathfrak q^{sh}$ is flat, the tor amplitude of this as an $R$-module is the same as the tor amplitude of $K_\mathfrak q$ as an $R$-module (More on Algebra, Lemma \ref{more-algebra-lemma-no-change-tor-amplitude}). Also, $K_{\mathfrak q}$ is a localization of $K_{\mathfrak q_0}$. Hence it suffices to show that $K_{\mathfrak q_0}$ has tor amplitude in $[a - d, b]$ as a complex of $R$-modules. \medskip\noindent Let $I \subset \mathfrak p_0 \subset R$ be the prime ideal corresponding to $f(x_0)$. Then we have \begin{align*} K \otimes_R^\mathbf{L} \kappa(\mathfrak p_0) & = (K \otimes_R^\mathbf{L} R/I) \otimes_{R/I}^\mathbf{L} \kappa(\mathfrak p_0) \\ & = (K \otimes_A^\mathbf{L} A/IA) \otimes_{R/I}^\mathbf{L} \kappa(\mathfrak p_0) \end{align*} the second equality because $R \to A$ is flat. By our choice of $a, b$ this complex has cohomology only in degrees in the interval $[a, b]$. Thus we may finally apply More on Algebra, Lemma \ref{more-algebra-lemma-lift-from-fibre-relatively-perfect} to $R \to A$, $\mathfrak q_0$, $\mathfrak p_0$ and $K$ to conclude.