Suppose $f:X \to Y$ is a finite morphism with $X$ and $Y$ being affine varieties, such that $X$ is unirational. In fact $X$ is more than unirational, it is the image of a morphism from a zariski open subset of $\mathbb P^n$. Are there any results that allows one to conclude the finite generation of the Chow ring $A^{*}(X) \otimes \mathbb Q$, if it is known that the chow ring $A^{*}(Y) \otimes \mathbb Q$ is finitely generated, are there any counter examples to this assertion?

I am assuming that the varieties are defined over $\mathbb C$.