  Let $X$ be a smooth projective variety with a fibration into varieties that
either satisfy a condition on representability of zero-cycles or that are
torsors under an abelian variety. We study the classes in the Brauer group that
never obstruct the Hasse principle for $X$. We prove that if the generic fiber
has a zero-cycle of degree $d$ over the generic point, then the Brauer classes
whose orders are prime to $d$ do not play a role in the Brauer--Manin
obstruction. As a result we show that the odd torsion Brauer classes never
obstruct the Hasse principle for del Pezzo surfaces of degree 2, certain K3
surfaces, and Kummer varieties.
