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) that induce isomorphisms on the $Chow_0$-groups ($0$-cycles modulo rational equivalence), and possibly also on $Chow_1$, and/or also on some 'higher Chow groups of low dimension'?