This is easy to answer in equicharacteristic under the additional assumption that $f$ is finite flat and $X$ and $Y$ are irreducible and generically reduced, using $\mathbf Z[\tfrac{1}{p}]$-coefficients, where $p$ is the exponential characteristic (i.e. $1$ if we're in characteristic $0$).

Indeed, in this case push and pull shows that the composition
$$\operatorname{CH}_*(Y) \stackrel{f^*}\to \operatorname{CH}_*(X) \stackrel{f_*}\to \operatorname{CH}_*(Y)$$
equals the degree of $f$ (see e.g. Fulton, Example 1.7.4). Since $f$ is radicial, the extension $K(Y) \to K(X)$ has degree $p^n$ for some $n \in \mathbf N$, hence the same goes for $f$. Thus $f^*$ is injective with $\mathbf Z[\tfrac{1}{p}]$-coefficients.

Similarly, if $V \subseteq Y$ is an integral subscheme with image $W$, then $f_*[V] = p^r[W]$ for some $r \in \mathbf N$, and $f^*[W] = m[V]$ for some $m \in \mathbf Z_{>0}$. Pushing forward gives
$$p^n[W] = f_*f^*[W] = mp^r[W],$$
so $m$ is a power of $p$ as well (or $[W] = 0$ hence $[V] = 0$), showing surjectivity with $\mathbf Z[\tfrac{1}{p}]$-coefficients. $\square$

This is enough to go from $k$ to $k^{\operatorname{perf}}$ with $\mathbf Z[\tfrac{1}{p}]$-coefficients by a limit argument (see e.g. Tag [0FH6](https://stacks.math.columbia.edu/tag/0FH6)).

**Example.** Injectivity is not true integrally, even for divisors on smooth projective $\bar{\mathbf F}_p$-varieties. For example, consider a $\pmb\mu_p$-quotient $X \to Y$ of smooth projective varieties with $X$ a complete intersection of dimension $\geq 2$. Then $\mathbf{Pic}^\tau_X = 0$ but $\mathbf{Pic}^\tau_Y = (\pmb\mu_p)^\vee = \mathbf Z/p$. See for example Cor. 1.2 of [this preprint of mine](https://arxiv.org/abs/2001.02787) for a modern account (but the result is much older).

I'm not sure if the same phenomenon occurs for a base change along a purely inseparable field extension.

For surjectivity with $\mathbf Z$-coefficients, by the argument above, it suffices to move a subscheme to a linearly equivalent one for which $[K(V):K(W)] = [K(X):K(Y)]$. This seems plausible at least in the quasi-projective case.

**Remark.** If you don't assume $f$ is flat or l.c.i. or $Y$ is regular, I'm not sure what the definition of $f^*$ is. (At least Fulton does not define this, nor does the Stacks project as far as I can tell.)