  We study zero-cycles in families of rationally connected varieties. We show
that for a smooth projective scheme over a henselian discrete valuation ring
the restriction of relative zero cycles to the special fiber induces an
isomorphism on Chow groups if the special fiber is separably rationally
connected. We further extend this result to certain higher Chow groups and
develop conjectures in the non-smooth case. Our main results generalise a
result of Koll\'ar [31].
