Which field extensions do not affect Chow groups?

Let $X$ be a (say, smooth projective) variety over a field $k$. For which $K$ it is known that the ("ordinary", that is, not higher) Chow groups of $X$ map onto that of $X_K$ bijectively?

This statement appears to be easy if $K$ is stably rational over $k$. What about other extensions; any references on this matter?