It is well-known (and was proved by Gabber?): if $R$ is a regular henselian local ring containing a field of characteristic prime to $l$, $k$ is its residue field, then $K_\ast(R,\mathbb{Z}/l)\cong K_\ast(k,\mathbb{Z}/l)$. My question is: are there any more classes of (regular) local rings such that this is true for them? Conversely, for which types of local rings this statement is 'usually' wrong?