  For a variety over a global field, one can consider subsets of the set of
adelic points of the variety cut out by finite abelian descent or Brauer-Manin
obstructions. Given a Galois extension of the ground field one can consider
similar sets over the extension and take Galois invariants. In this paper, we
study under which circumstances the Galois invariants recover the obstruction
sets over the ground field. As an application of our results, we study finite
abelian descent and Brauer-Manin obstructions for isotrivial curves over
function fields and extend results obtained by the first and last authors for
constant curves to the isotrivial case.
