  We study the domination monoid in various classes of structures arising from
the model theory of henselian valuations, including RV-expansions of henselian
valued fields of residue characteristic 0 (and, more generally, of benign
valued fields), p-adically closed fields, monotone D-henselian differential
valued fields with many constants, regular ordered abelian groups, and pure
short exact sequences of abelian structures. We obtain Ax-Kochen-Ershov type
reductions to suitable fully embedded families of sorts in quite general
settings, and full computations in concrete ones.
