A reference for birational equivalence of $CH_0$ is Fulton's *Intersection Theory* [1], **Example 16.1.11**. In the example, he makes the assumption that $k$ is algebraically closed, but he never uses it. Since the argument is fairly short, let me repeat it here.

**Theorem.** Let $k$ be a field, and let $X$ and $Y$ be smooth proper $k$-varieties. If $X$ and $Y$ are birational, then $CH_0(X) \cong CH_0(Y)$.

*Proof.* Let $f \colon X \dashrightarrow Y$ be a birational map. Let $\Gamma \subseteq X \times Y$ be the closure of the graph. Then $\Gamma$ defines maps
\begin{align*}
f_* \colon CH_0(X) &\to CH_0(Y) & f^* \colon CH_0(Y) &\to CH_0(X)\\
a &\mapsto \pi_{Y,*}(\Gamma \cdot \pi_X^* a), & b &\mapsto \pi_{X,*}(\Gamma \cdot \pi_Y^* b).
\end{align*}
The composition $f^* \circ f_*$ (resp. $f_* \circ f^*$) is given by the cycle $\Gamma^\top \circ \Gamma := \pi_{13, *} (\pi_{12}^* \Gamma \cdot \pi_{23}^* \Gamma^\top)$ on $X \times X$ (resp. by $\Gamma \circ \Gamma^\top$ on $Y \times Y$); see [*loc. cit.*, **Def. 16.1.1** and **Prop. 16.1.2(a)**] for details.

Let $V \subseteq Y$ be an open such that $f$ induces an isomorphism $f^{-1}(V) \stackrel\sim\to V$. Let $U = f^{-1}(V)$, $Z = X \setminus U$, and $W = Y \setminus V$. I claim that the cycle $\varepsilon := \Gamma^\top \circ \Gamma - \Delta_X$ is supported on $Z \times Z$. By the short exact sequence of [*loc. cit.*, **Prop. 1.8**], it suffices to show that the restriction of $\varepsilon$ to $S := X \times X \setminus Z \times Z = X \times U \cup U \times X$ is zero. Let
$$T := (X \times Y \times U) \cup (U \times Y \times X) = \pi_{13}^{-1} (S).$$
Then the restriction of $\varepsilon$ to $S$ is the pushforward along $\pi_{13}$ of
$$\left.\left(\pi_{12}^* \Gamma \cdot \pi_{23}^* \Gamma^\top\right)\right|_T.\tag{1}$$
We can compute the latter as
$$\left.\left(\pi_{12}^* \Gamma\right)\right|_T \cdot \left.\left(\pi_{23}^* \Gamma^\top\right)\right|_T.$$
This is a proper intersection, and the intersection is equal to the 'diagonal'
$$\left\{(a,b,c) \in U \times V \times U\ \big|\ f(a)=b=f(c)\right\}.$$
Indeed, the intersection agrees with this set on both $X \times Y \times U$ and $U \times Y \times X$.

Then the pushforward of (1) is $\Delta_U$. Hence $\varepsilon$ vanishes on $S$, so it is supported on $Z \times Z$. In particular, the projections $\pi_{1,*} \varepsilon, \pi_{2,*} \varepsilon \in CH_*(X)$ are supported on $Z$.

The punchline is that $\varepsilon$ acts as the identity on $CH_0(X)$ by the moving lemma. Indeed, any $0$-cycle $a$ on a smooth variety can be moved away from $Z$, so the intersection $\pi_1^* a \cdot \varepsilon$ is zero.

This proves that $f^* \circ f_*$ is the identity, and by symmetry the same holds for $f_* \circ f^*$. $\square$

**Remark.** It is even true that $CH_0$ is a stable birational invariant: if $X$ and $Y$ are smooth and proper varieties with $X \times \mathbb P^n \stackrel\sim\dashrightarrow Y \times \mathbb P^m$, then $CH_0(X) \cong CH_0(Y)$. (The only thing left to prove is that $CH_0(X) \cong CH_0(X \times \mathbb P^1)$, since we already know the result for birational varieties.)

**Remark.** The reason Fulton only proves the theorem for algebraically closed fields is that Fulton only proves the moving lemma over those fields. For a proof of the moving lemma over arbitrary fields, see Roberts's appendix to the Oslo 1970 Algebraic Geometry proceedings [2] (finite fields are addressed separately).

**References:**

[1] **Fulton, William**. Intersection theory (second edition). Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. *Springer-Verlag*, Berlin, 1998. ISBN: 3-540-62046-X; 0-387-98549-2. MR1644323

[2] **Roberts, Joel**. Chow's moving lemma. Appendix 2 to: "Motives" by Steven L. Kleiman. *Algebraic geometry, Oslo 1970 (Proc. Fifth Nordic Summer School in Math.)*, pp. 89--96. *Wolters-Noordhoff*, Groningen, 1972. MR0382269