Case (\ref{item-representable}) follows from Lemma \ref{lemma-thicken-property-morphisms}. \medskip\noindent Choose a scheme $U'$ and a surjective \'etale morphism $U' \to B'$. Choose a scheme $V'$ and a surjective \'etale morphism $V' \to U' \times_{B'} Y'$. Choose a scheme $W'$ and a surjective \'etale morphism $W' \to V' \times_{Y'} X'$. Let $U, V, W$ be the base change of $U', V', W'$ by $B \to B'$. Consider the diagram $$ \xymatrix{ (W \subset W') \ar[rr] \ar[rd] & & (V \subset V') \ar[ld] \\ & (U \subset U') } $$ of thickenings of schemes. For any of the properties which are \'etale local on the source-and-target the result follows immediately from the corresponding result for morphisms of thickenings of schemes applied to the diagram above. Thus cases (\ref{item-flat}), (\ref{item-finite-type}), (\ref{item-quasi-finite}), (\ref{item-finite-presentation}), (\ref{item-relative-dimension-d}), (\ref{item-syntomic}), (\ref{item-smooth}), (\ref{item-unramified}), (\ref{item-etale}) follow from the corresponding cases of More on Morphisms, Lemma \ref{more-morphisms-lemma-deform-property}. \medskip\noindent Since $X \to X'$ and $Y \to Y'$ are universal homeomorphisms we see that any question about the topology of the maps $X \to Y$ and $X' \to Y'$ has the same answer. Thus we see that cases (\ref{item-quasi-compact}), (\ref{item-universally-closed}), (\ref{item-surjective}), (\ref{item-universally-injective}), and (\ref{item-universally-open}) hold. \medskip\noindent In each of the remaining cases we only prove the implication $f\text{ has }P \Rightarrow f'\text{ has }P$ since the other implication follows from the fact that $P$ is stable under base change, see Spaces, Lemma \ref{spaces-lemma-base-change-immersions} and Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-base-change-separated}, \ref{spaces-morphisms-lemma-base-change-monomorphism}, \ref{spaces-morphisms-lemma-base-change-affine}, \ref{spaces-morphisms-lemma-base-change-proper}, \ref{spaces-morphisms-lemma-base-change-integral}, and \ref{spaces-morphisms-lemma-base-change-finite-locally-free}. \medskip\noindent The case (\ref{item-open-immersion}). Assume $f$ is an open immersion. Then $f'$ is \'etale by (\ref{item-etale}) and universally injective by (\ref{item-universally-injective}) hence $f'$ is an open immersion, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-etale-universally-injective-open}. You can avoid using this lemma at the cost of first using (\ref{item-representable}) to reduce to the case of schemes. \medskip\noindent The case (\ref{item-isomorphism}). Follows from cases (\ref{item-open-immersion}) and (\ref{item-surjective}). \medskip\noindent The case (\ref{item-separated}). See Lemma \ref{lemma-thicken-property-morphisms}. \medskip\noindent The case (\ref{item-monomorphism}). Assume $f$ is a monomorphism. Consider the diagonal morphism $\Delta_{X'/Y'} : X' \to X' \times_{Y'} X'$. The base change of $\Delta_{X'/Y'}$ by $B \to B'$ is $\Delta_{X/Y}$ which is an isomorphism by assumption. By (\ref{item-isomorphism}) we conclude that $\Delta_{X'/Y'}$ is an isomorphism and hence $f'$ is a monomorphism. \medskip\noindent The case (\ref{item-affine}). See Lemma \ref{lemma-thicken-property-morphisms}. \medskip\noindent The case (\ref{item-proper}). See Lemma \ref{lemma-thicken-property-morphisms-cartesian}. \medskip\noindent The case (\ref{item-integral}). See Lemma \ref{lemma-thicken-property-morphisms}. \medskip\noindent The case (\ref{item-finite}). See Lemma \ref{lemma-thicken-property-morphisms-cartesian}. \medskip\noindent The case (\ref{item-finite-locally-free}). Assume $f$ finite locally free. By (\ref{item-finite}) we see that $f'$ is finite. By (\ref{item-flat}) we see that $f'$ is flat. By (\ref{item-finite-presentation}) $f'$ is locally of finite presentation. Hence $f'$ is finite locally free by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-flat}.