The domination monoid in henselian valued fields

We study the domination monoid in various classes of structures arising from henselian valuations, including RV-expansions of henselian valued fields of equicharacteristic 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.