Are birational morphisms stable under base change via a dominant morphism Let $f: X \to Y$ be a birational morphism of integral schemes and $g: Z \to Y$ a morphism of integral schemes which maps the generic point of $Z$ to the generic point of $Y$, i.e., the morphism $g$ is dominant.
Is then $X \times_Y Z \to Z$ birational?
Edit: My ideas: Denote the generic points of $X,Y,Z$ by $\eta_X, \eta_Y, \eta_Z$. Then $f$ induces an isomorphism $\eta_X = \eta_Y$. Denote the base change $X \times_Y Z \to Z$ by $f'$. Then $f'$ induces an isomorphism $g'^*(\eta_X) = \eta_Z$?

Since $f: X \to Y$ is birational, we can find some open subsets $U \subset Y$ and $V \subset X$ so that $f$ restricts to an isomorphism $f: V \to U$. Then $g: W = g^{-1}(U) \to U$ is still dominant (really dominance of $g$ here just guarantees that $W$ is nonempty for any open $U$ we may need to restrict to). 
Then the pullback $V \times_U W \to W$ is an isomorphism. $V \times_U W$ is an open subset of $X \times_Y Z$ and the morphism is just the restriction of the pullback $X \times_Y Z \to Z$. Thus the pullback is birational since it induces an isomorphism on open subsets.
