TITLE: Reference request on birational invariance of Chow group of zero cycles of degree zero
QUESTION [9 upvotes]: Let $CH_0(X)^0$ denote the group of zero cycles of degree zero modulo rational equivalence.
I am looking for a reference for the following fact:
If $X$ and $Y$ are smooth and projective varieties over a finite field and we suppose they are birational, then the groups $CH_0(X)^0$ and $CH_0(Y)^0$ are isomorphic.
As a second question, let $k$ denote the base field and set
$$
\pi_1^{ab}(X)^0 := \text{ker}[\pi_1^{ab}(X) \rightarrow \pi_1^{ab}(k)].
$$
Then, with the same assumptions on $X$ and $Y$, $\pi_1^{ab}(X)^0$ and $\pi_1^{ab}(Y)^0$ should be isomorphic.
For the first fact, I have found a reference for the case of dimension 2 in the answer to this question (which links to a 1979 paper of Colliot-Thélène and Coray). The cases of dimension 1 and 0 are obvious. However, I have failed to find a reference for the case of general dimension. Any help is greatly appreciated.

REPLY [3 votes]: Using Jason Starr's comment I was able (I think) to figure out the case of $\pi_1(X)^0$. For anyone who stumbles across this with the same question in mind I add a sketch of the proof in an answer. For the experts, if you feel like leaving a comment whether my reasoning is correct, that would be great.
Let $\phi: X \rightarrow Y$ be a rational map, i.e. defined on some open $U \subset X$. Because $X$ is normal, $Z := X \backslash U$ has codimension at least two. This means the dimenson of the ring $\mathcal{O}_{X,z}$ is at least two for each point $z \in Z$. But then this local ring is pure (SGA 2, Exp. X, Thm 3.4), which in turn implies the couple $(X,Z)$ is pure (SGA 2, Exp. X, prop. 3.3), so that the categories $\operatorname{FEt}(X)$ and $\operatorname{FEt}(U)$ are equivalent.
Since $X$ and $Y$ are birational, we can find an open $V \subset Y$ such that $U$ is isomorphic to $V$ and in the same way we show $V$ and $Y$ have the same \'etale coverings. This shows $\pi_1(X) = \pi_1(U) = \pi_1(V) = \pi_1(Y)$ as desired, which proves in particular what I wanted to show.