Choose a scheme $U$ and a surjective smooth morphism $U \to \mathcal{X}$. We first reduce the lemma to the case where $x$ lifts to $U$. Recall that $T_x(M)$ is the set of isomorphism classes of lifts of $x$ to $A[M]$. Therefore Lemma \ref{lemma-sheaf-of-infinitesimal-lifts}\footnote{This lemma applies: $\Delta : \mathcal{X} \to \mathcal{X} \times_S \mathcal{X}$ is locally of finite presentation by Morphisms of Stacks, Lemma \ref{stacks-morphisms-lemma-finite-presentation-permanence} and the assumption that $\mathcal{X} \to S$ is locally of finite presentation. Therefore $\mathcal{I}_\mathcal{X} \to \mathcal{X}$ is locally of finite presentation as a base change of $\Delta$.} says that the rule $$ A_1 \mapsto T_{x|_{A_1}}(M \otimes_A A_1) $$ is a sheaf on the small \'etale site of $\Spec(A)$; the tensor product is needed to make $A[M] \to A_1[M \otimes_A A_1]$ a flat ring map. We may choose a faithfully flat \'etale ring map $A \to A_1$ such that $x|_{A_1}$ lifts to a morphism $u_1 : \Spec(A_1) \to U$, see for example Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-surjective-flat-locally-finite-presentation}. Write $A_2 = A_1 \otimes_A A_1$ and set $B_1 = B \otimes_A A_1$ and $B_2 = B \otimes_A A_2$. Consider the diagram $$ \xymatrix{ 0 \ar[r] & T_y(M \otimes_A B) \ar[r] & T_{y|_{B_1}}(M \otimes_A B_1) \ar[r] & T_{y|_{B_2}}(M \otimes_A B_2) \\ 0 \ar[r] & T_x(M) \ar[r] \ar[u] & T_{x|_{A_1}}(M \otimes_A A_1) \ar[r] \ar[u] & T_{x|_{A_2}}(M \otimes_A A_2) \ar[u] } $$ The rows are exact by the sheaf condition. We have $M \otimes_A B_i = (M \otimes_A A_i) \otimes_{A_i} B_i$. Thus if we prove the result for the middle and right vertical arrow, then the result follows. This reduces us to the case discussed in the next paragraph. \medskip\noindent Assume that $x$ is the image of a morphism $u : \Spec(A) \to U$. Observe that $T_u(M) \to T_x(M)$ is surjective since $U \to \mathcal{X}$ is smooth and representable by algebraic spaces, see Criteria for Representability, Lemma \ref{criteria-lemma-representable-by-spaces-formally-smooth} (see discussion preceding it for explanation) and More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-smooth-formally-smooth}. Set $R = U \times_\mathcal{X} U$. Recall that we obtain a groupoid $(U, R, s, t, c, e, i)$ in algebraic spaces with $\mathcal{X} = [U/R]$. By Artin's Axioms, Lemma \ref{artin-lemma-ses-inf-and-T} we have an exact sequence $$ T_{e \circ u}(M) \to T_u(M) \oplus T_u(M) \to T_x(M) \to 0 $$ where the zero on the right was shown above. A similar sequence holds for the base change to $B$. Thus the result we want follows if we can prove the result of the lemma for $T_u(M)$ and $T_{e \circ u}(M)$. This reduces us to the case discussed in the next paragraph. \medskip\noindent Assume that $\mathcal{X} = X$ is an algebraic space locally of finite presentation over $S$. Then we have $$ T_x(M) = \Hom_A(x^*\Omega_{X/S}, M) $$ by the discussion in More on Morphisms of Spaces, Section \ref{spaces-more-morphisms-section-action-by-derivations}. By the same token $$ T_y(M \otimes_A B) = \Hom_B(y^*\Omega_{X/S}, M \otimes_A B) $$ Since $X \to S$ is locally of finite presentation, we see that $\Omega_{X/S}$ is locally of finite presentation, see More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-finite-presentation-differentials}. Hence $x^*\Omega_{X/S}$ is a finitely presented $A$-module. Clearly, we have $y^*\Omega_{X/S} = x^*\Omega_{X/S} \otimes_A B$. we conclude by More on Algebra, Lemma \ref{more-algebra-lemma-pseudo-coherence-and-base-change-ext}.