Let $X$, $Y$, $Z$ be integral schemes of finite type over a field $K$ (i.e., locally affine opens are finitely-generated algebras over $K$). Suppose we have the following condition$\colon$

$f \colon Y \to X$ is finite.

For a dominant morphism $g \colon Z \to X$, suppose that there is NO dominant morphism $Z \to Y$ between schemes. (That is, there is NO embedding ${\cal O}_Y \hookrightarrow {\cal O}_Z$ between structural rings).

Q. Is it possble that $Y \times_X Z$ is not integral while $Y \times_X \eta_Z$ being integral ?

As in the answer below, in the case $Z = Y$, there is an example by normalisation of $X$.