This question is a contination 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 isomorphism, and possibly the same is true for on $Chow_1$, and/or also for some 'higher Chow groups of low dimension'? Currently I am only aware of papers that treat embeddings of varieties (composed with birational morphisms).