  We study the Brauer-Manin obstruction to the existence of zero-cycles of
degree $d$ on certain classes of varieties over number fields. We generalise
existing results in the literature and prove some results about fibrations over
the projective line, where the geometric Brauer group of the generic fibre is
not assumed to be finite. The idea is to assume that the Brauer-Manin
obstruction to the Hasse principle is the only one for certain fibres and then
deduce analogous results for zero-cycles.
