  In this paper, we study weak approximation with Brauer--Manin obstruction
with respect to extensions of number fields. For any nontrivial extension
$L/K,$ assuming a conjecture of M. Stoll, we prove that there exists a
$K$-threefold satisfying weak approximation with Brauer--Manin obstruction off
all archimedean places, while its base change to $L$ fails. Then we illustrate
this construction with an explicit unconditional example.
