I am wondering whether something like the following argument is correct or not?
If a morphism of regular schemes $f:X\rightarrow Y$ induces an isomorphism on residue fields and also induces a bijection between closed points of $X$ and $Y$ then it induces an isomorphism $K_n^{ét}(X,\mathbb{Z}/l)\cong K_n^{ét}(Y,\mathbb{Z}/l)$.
Proof: In order to prove that $f$ induces an isomorphism of etale $K$-theory it suffices to prove it locally. Since etale $K$-theory is sheaf of spectra on the etale site, then it will be enough to show this map induces homotopy equivalence on the etale $K$-theory of the strict henselization of closed points. Assume $f$ maps a closed point $p$ to $f(p)$ and $\kappa_p$, $\kappa_{f(p)}$ are residue fields that $f$ induces an isomorphism between them. After strict henselization at the closed points we have pair $(A, \kappa_p^{sep})$ and $(B, \kappa_{f(p)}^{sep})$, where these are the local rings at the source and target, with the corresponding residue fields. The map $f$ should induce isomorphism on the residue fields after the strict heneselization. Since we are working locally etale $K$-theory should coincide with Zariski $K$-theory. Because everything is regular there are no negative $K$-groups we just need to prove that $f$ induces isomoprhism between $K_n^{ét}(A,\mathbb{Z}/l)$ and $K_n^{ét}(B,\mathbb{Z}/l)$ for $n\geq 0$. By Gabber's rigidity this reduces to showing the isormphism at the level of residue fields. But we know on the residue fields this is an isomorphism.