  This paper focuses on flow-adapted point-shifts of point processes on
topological groups, which map points of a point process to other points of the
point process in a translation invariant way. Foliations and connected
components generated by point-shifts are studied, and the cardinality
classification of connected components, previously known on Euclidean space, is
generalized to unimodular groups. An explicit counterexample is also given on a
non-unimodular group. Isomodularity of a point-shift is defined and identified
as a key component in generalizations of Mecke's invariance theorem in the
unimodular and non-unimodular cases. Isomodularity is also the deciding factor
of when the reciprocal and reverse of a point-map corresponding to a bijective
point-shift are equal in distribution. Next, sufficient conditions for
separating points of a point process are given. Finally, connections between
point-shifts of point processes and vertex-shifts of unimodular networks are
given that allude to a deeper connection between the theories.
