ag.algebraic geometry - Example of restriction of a finite morphism which is not finite

Every closed immersion is a finite morphism. Therefore, restriction of a finite morphism to a closed subset is always a finite morphism itself. Can you give an example of quasi-projective varieties $Xsubset Y$, $Z$ and a finite morphism $f:Yto Z$ such that restriction $f:Xto f(X)$ is not finite? Same with Y -- projective?

PS. Sorry the original version of this question was hilariously stupid.