  Conjectures on the existence of zero-cycles on arbitrary smooth projective
varieties over number fields were proposed by Colliot-Th\'el\`ene, Sansuc, Kato
and Saito in the 1980's. We prove that these conjectures are compatible with
fibrations, for fibrations into rationally connected varieties over a curve. In
particular, they hold for the total space of families of homogeneous spaces of
linear groups with connected geometric stabilisers. We prove the analogous
result for rational points, conditionally on a conjecture on locally split
values of polynomials which a recent work of Matthiesen establishes in the case
of linear polynomials over the rationals.
