Local-global principle for 0-cycles on fibrations over rationally connected bases

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.