Setting $X' = \coprod X'_i$ we see that (2) implies (1). Write $X = \Spec(A)$ and $Y = \Spec(B)$. Write $A$ as a filtered colimit of finite type $\mathbf{Z}$-algebras $A_i$. Since $B$ is an $A$-algebra of finite presentation, we see that there exists $0 \in I$ and a finite type ring map $A_0 \to B_0$ such that $B = \colim B_i$ with $B_i = A_i \otimes_{A_0} B_0$, see Algebra, Lemma \ref{algebra-lemma-colimit-category-fp-algebras}. For $i$ sufficiently large we see that $A_i \to B_i$ is quasi-finite, see Limits, Lemma \ref{limits-lemma-descend-quasi-finite}. Thus we reduce to the case of finite type algebras over $\mathbf{Z}$, in particular we reduce to the Noetherian case. (Details omitted.) \medskip\noindent Assume $X$ and $Y$ Noetherian. In this case any locally closed subset of $X$ is constructible. By Lemma \ref{lemma-generically-finite} and Noetherian induction we see that there is a finite partition $X = \coprod X_i$ of $X$ by locally closed strata such that $Y \times_X X_i \to X_i$ is finite. We can refine this partition to get affine strata. Thus after replacing $X$ by $X' = \coprod X_i$ we may assume $Y \to X$ is finite. \medskip\noindent Assume $X$ and $Y$ Noetherian and $Y \to X$ finite. Suppose that we can prove (2) after base change by a surjective, flat, quasi-finite morphism $U \to X$. Thus we have a partition $U = \coprod U_i$ and finite locally free morphisms $U'_i \to U_i$ such that $U'_i \times_X Y \to U'_i$ is isomorphic to $\coprod_{j = 1}^{n_i} (U'_i)_{red} \to (U'_i)_{red}$ for some $n_i$. Then, by the argument in the previous paragraph, we can find a partition $X = \coprod X_j$ with locally closed affine strata such that $X_j \times_X U_i \to X_j$ is finite for all $i, j$. By Morphisms, Lemma \ref{morphisms-lemma-finite-flat} each $X_j \times_X U_i \to X_j$ is finite locally free. Hence $X_j \times_X U'_i \to X_j$ is finite locally free (Morphisms, Lemma \ref{morphisms-lemma-composition-finite-locally-free}). It follows that $X = \coprod X_j$ and $X_j' = \coprod_i X_j \times_X U'_i$ is a solution for $Y \to X$. Thus it suffices to prove the result (in the Noetherian case) after a surjective flat quasi-finite base change. \medskip\noindent Applying Morphisms, Lemma \ref{morphisms-lemma-massage-finite} we see we may assume that $Y$ is a closed subscheme of an affine scheme $Z$ which is (set theoretically) a finite union $Z = \bigcup_{i \in I} Z_i$ of closed subschemes mapping isomorphically to $X$. In this case we will find a finite partition of $X = \coprod X_j$ with affine locally closed strata that works (in other words $X'_j = X_j$). Set $T_i = Y \cap Z_i$. This is a closed subscheme of $X$. As $X$ is Noetherian we can find a finite partition of $X = \coprod X_j$ by affine locally closed subschemes, such that each $X_j \times_X T_i$ is (set theoretically) a union of strata $X_j \times_X Z_i$. Replacing $X$ by $X_j$ we see that we may assume $I = I_1 \amalg I_2$ with $Z_i \subset Y$ for $i \in I_1$ and $Z_i \cap Y = \emptyset$ for $i \in I_2$. Replacing $Z$ by $\bigcup_{i \in I_1} Z_i$ we see that we may assume $Y = Z$. Finally, we can replace $X$ again by the members of a partition as above such that for every $i, i' \subset I$ the intersection $Z_i \cap Z_{i'}$ is either empty or (set theoretically) equal to $Z_i$ and $Z_{i'}$. This clearly means that $Y$ is (set theoretically) equal to a disjoint union of the $Z_i$ which is what we wanted to show.