Briefly, which properties morphism should have to induce pushforward of Chow rings? It is said in both Hartshorne and 3264 and All That that it should be proper, but why not just closed?
Let $Z(X)$ be cycles on an algebraic variety $X$, i.e. linear combinations of (closed) subvarieties, and $Rat(X) \subset Z(X)$ be subgroup generated by $[S \cap \{0\} \times X]-[S \cap \{1\} \times X]$ for irreducible subvarieties $S \subset \mathbb P^1\times X$, not lying in one layer of this product. Then $A(X)=Z(X)/Rat(X)$ is Chow group, and may be equipped with ring structure.
Let $f: X\to Y$ be a closed morphism of smooth varieties. Then one can pushforward cycle $Z$ on $X$ to either 0 if $\dim f(X)<\dim X$ or to $[K(Z):K(f(Z))] f(Z)$ otherwise. It seems clearly that for closed $f$ $Rat(X)$ will be sent to $Rat(Y)$, am I wrong?