  This paper proves a number of flatness results for centralizers of sections
of a reductive group scheme over a general base scheme. To this end, we
establish relative versions of the Jordan decomposition. Using our results, we
obtain a canonical flattening stratification for the universal centralizer of a
simply connected semisimple group scheme over a base of good characteristic. We
also investigate the structure of centralizers and conjugacy classes of
unipotent and nilpotent sections over general bases.
