This question is a continuation of Varieties with Chow groups supported in positive codimension: examples and properties?

What examples are known of morphisms of varieties and Chow motives (say, over complex numbers) such that the pushforward map on $Chow_0$ (the group of $0$-cycles modulo rational equivalence) is an epimorphism or an isomorphism, and possibly the same is true for on $Chow_1$? I would like to have an example that is rather "compicated", so that (1) one cannot compute the groups in question themselves and (2) the isomorphism in question cannot be established using a "simple" general (motivic or intersection theory) argument.