  We study the Brauer-Manin obstruction to the Hasse principle and to weak
approximation for 0-cycles on algebraic varieties that possess a fibration
structure. The exactness of the local-to-global sequence $(E)$ of Chow groups
of 0-cycles was known only for a fibration whose base is either a curve or the
projective space. In the present paper, we prove the exactness of $(E)$ for
fibrations whose bases are Ch\^{a}telet surfaces or projective models of
homogeneous spaces of connected linear algebraic groups with connected
stabilizers. We require that either all fibres are split and most fibres
satisfy weak approximation for 0-cycles, or the generic fibre has a 0-cycle of
degree $1$ and $(E)$ is exact for most fibres.
