Chow group and base change Let $k$ be a field with algebraic closure $\overline{k}$. Let $f\colon X\to k$ be a smooth projective variety(geometrically connected) over $k$. 
Is the base change map $$\phi_i\colon \mathrm{CH}^i(X)\to\mathrm{CH}^{i}(X_{\overline{k}})$$ always injective?
(If $i=1$, $\mathrm{CH}^1(X)=\mathrm{Pic}(X)$, the Hochschild-Serre spectral sequence gives $$0\to H^1(\mathrm{Gal}(\overline{k}/k),f_{\overline{k},*}\mathbb{G}_m)\to\mathrm{Pic}(X)\to\mathrm{Pic}_{X_\overline{k}/\overline{k}}(k)\to H^2(\mathrm{Gal}(\overline{k}/k),f_{\overline{k},*}\mathbb{G}_m)$$
The condition implies $f_{\overline{k},*}\mathbb{G}_m=\mathbb{G}_m$, so $H^1(\mathrm{Gal}(\overline{k}/k),f_{\overline{k},*}\mathbb{G}_m)=0$, and $\mathrm{Pic}_{X_{\overline{k}}/\overline{k}}(k)\to\mathrm{Pic}_{X_{\overline{k}}/\overline{k}}(\overline{k})$ is injective, we know $\phi_1$ is injective. 
If $X$ is quasi-projective, then $\phi_1$ is not injective, for example $X=\mathbb{P}^1_{\mathbb{R}}-\{\pm i\}$, then $\mathcal{O}(1)$ is a nontrivial element in $\mathrm{ker}(\phi_1)$.) 
 A: No, this is not true in general.
A counterexample occurs already for Severi-Brauer varieties. Since the Chow group $\text{CH}(\mathbf{P}^n)$ is torsion free, it's enough to show there are Severi-Brauer varieties with torsion in their Chow groups. This was (I think) first observed in:

Merkurjev, A. S. Certain K-cohomology groups of Severi-Brauer varieties. K-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), 319–331, Proc. Sympos. Pure Math., 58, Part 2, Amer. Math. Soc., Providence, RI, 1995.

Merkurjev uses the BGQ spectral sequence to show that certain differentials that compute the Chow groups are nonzero. Since it's also known from the Grothendieck-Riemann-Roch without denominators that the image of these differentials is contained in the torsion subgroup, this proves the claim.
More precise results in this direction appeared in:

Karpenko, Nikita A. Codimension 2 cycles on Severi-Brauer varieties. K-Theory 13 (1998), no. 4, 305–330. (Available at the authors website https://sites.ualberta.ca/~karpenko/publ/ch2.pdf).

Karpenko uses the gamma filtration on the Grothendieck ring of Severi-Brauer varieties to explicitly construct torsion elements in $\text{gr}_\gamma^2K(-)$ for certain Severi-Brauer varieties. He then shows this is equal to $\text{CH}^2(-)$ for Severi-Brauer varieties of certain indecomposable algebras.
More recently, Karpenko has provided a complete description of the Chow group of certain Severi-Brauer varieties in:

Karpenko, Nikita A.(3-AB-MS)
  Chow ring of generically twisted varieties of complete flags. (English summary) 
  Adv. Math. 306 (2017), 789–806.

A number of these have a lot of torsion in their Chow groups (see Examples 3.17-3.22).
