This question is a contination of http://mathoverflow.net/questions/182041/varieties-with-chow-groups-supported-in-positive-codimension-examples-and-prope

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'?