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, $g \colon Z \to X$ is dominant (i.e., there is an inclusion ${\cal O}_X \hookrightarrow {\cal O}_Z$).

Consider the fibre product $Y \times_X Z$. Let $\eta_Z \colon= {\mathrm{Spec}}\,k(Z)$ be a generic point of $Z$.

Q. Under the assumption

$k(Y) \cap k(Z) = k(X)$,

What is the example such that $Y \times_X Z$ is not integral ?