Choose a countable subfield $k \subset K$ for example the prime field. By Lemmas \ref{lemma-countable-finite-type} and \ref{lemma-countable-perfect} there exists a countable set of isomorphism classes of systems over $k$ satisfying parts (1), (2), (3) of the lemma. Thus we can choose a countable set $I$ and for each $i \in I$ such a system $$ (S_{0, i}/k, X_{0, i} \to S_{0, i}, Y_{0, i} \to S_{0, i}, M_{0, i}) $$ over $k$ such that each isomorphism class occurs at least once. Denote $(S_i/K, X_i \to S_i, Y_i \to S_i, M_i)$ the base change of the displayed system to $K$. This system has properties (1), (2), (3), see Lemma \ref{lemma-base-change-rek}. Let us prove property (4). \medskip\noindent Consider smooth proper schemes $X$ and $Y$ over $K$ such that there is a $K$-linear exact equivalence $F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$. By Proposition \ref{proposition-equivalence} we may assume that there exists an object $M \in D_{perf}(\mathcal{O}_{X \times Y})$ such that $F = \Phi_M$ is the corresponding Fourier-Mukai functor. By Lemma \ref{lemma-fourier-mukai-flat-proper-over-noetherian} there is an $M'$ in $D_{perf}(\mathcal{O}_{Y \times X})$ such that $\Phi_{M'}$ is the right adjoint to $\Phi_M$. Since $\Phi_M$ is an equivalence, this means that $\Phi_{M'}$ is the quasi-inverse to $\Phi_M$. By Lemma \ref{lemma-fourier-mukai-flat-proper-over-noetherian} we see that the Fourier-Mukai functors defined by the objects $$ A = R\text{pr}_{13, *}( L\text{pr}_{12}^*M \otimes_{\mathcal{O}_{X \times Y \times X}}^\mathbf{L} L\text{pr}_{23}^*M') $$ in $D_{perf}(\mathcal{O}_{X \times X})$ and $$ B = R\text{pr}_{13, *}( L\text{pr}_{12}^*M' \otimes_{\mathcal{O}_{Y \times X \times Y}}^\mathbf{L} L\text{pr}_{23}^*M) $$ in $D_{perf}(\mathcal{O}_{Y \times Y})$  are isomorphic to $\text{id} : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_X)$ and $\text{id} : D_{perf}(\mathcal{O}_Y) \to D_{perf}(\mathcal{O}_Y)$ Hence $A \cong \Delta_{X/K, *}\mathcal{O}_X$ and $B \cong \Delta_{Y/K, *}\mathcal{O}_Y$ by Lemma \ref{lemma-uniqueness}. Hence we see that $M$ is the Fourier-Mukai kernel of a relative equivalence from $X$ to $Y$ over $K$ by definition. \medskip\noindent We can write $K$ as the filtered colimit of its finite type $k$-subalgebras $A \subset K$. By Limits, Lemma \ref{limits-lemma-descend-finite-presentation} we can find $X_0, Y_0$ of finite type over $A$ whose base changes to $K$ produces $X$ and $Y$. By Limits, Lemmas \ref{limits-lemma-eventually-proper} and \ref{limits-lemma-descend-smooth} after enlarging $A$ we may assume $X_0$ and $Y_0$ are smooth and proper over $A$. By Lemma \ref{lemma-descend-rek} after enlarging $A$ we may assume $M$ is the pullback of some $M_0  \in D_{perf}(\mathcal{O}_{X_0 \times_{\Spec(A)} Y_0})$ which is the Fourier-Mukai kernel of a relative equivalence from $X_0$ to $Y_0$ over $\Spec(A)$. Thus we see that $(S_0/k, X_0 \to S_0, Y_0 \to S_0, M_0)$ is isomorphic to $(S_{0, i}/k, X_{0, i} \to S_{0, i}, Y_{0, i} \to S_{0, i}, M_{0, i})$ for some $i \in I$. Since $S_i = S_{0, i} \times_{\Spec(k)} \Spec(K)$ we conclude that (4) is true with $s : \Spec(K) \to S_i$ induced by the morphism $\Spec(K) \to \Spec(A) \cong S_{0, i}$ we get from $A \subset K$.