  We study the existence of zero-cycles of degree one on varieties that are
defined over a function field of a curve over a complete discretely valued
field. In particular, we show that local-global principles hold for such
zero-cycles provided that local-global principles hold for the existence of
rational points over extensions of the function field. This assertion is
analogous to a known result concerning varieties over number fields. We also
show that our results hold more generally in the henselian case.
