Suppose that $E/F$ is a unramified extension of local fields of characteristic zero. Let $G = GL_n$. Then it is well-known (due to Clozel?) that base change of tempered representations from $G(F)$ to $G(E)$ holds.
Question: does the same result hold in the case of characteristic $p > 0$?
EDIT: As Olivier says, this actually seems to follow immediately from LLC for function fields. Thanks!