  If dark matter (DM) interacts with the Standard Model (SM) via the kinetic
mixing (KM) portal, it necessitates the existence of portal matter (PM)
particles which carry both dark and SM quantum numbers that will appear in
vacuum polarization-like loop graphs. In addition to the familiar $\sim
e\epsilon Q$ strength, QED-like interaction for the dark photon (DP), in some
setups different loop graphs of these PM states can also induce other coupling
structures for the SM fermions that may come to dominate in at least some
regions of parameter space regions and which can take the form of `dark'
moments, eg, magnetic dipole-type interactions in the IR, associated with a
large mass scale, $\Lambda$. In this paper, motivated by a simple toy model, we
perform a phenomenological investigation of a possible loop-induced dark
magnetic dipole moment for SM fermions, in particular, for the electron. We
show that at the phenomenological level such a scenario can not only be made
compatible with existing experimental constraints for a significant range of
correlated values for $\Lambda$ and the dark $U(1)_D$ gauge coupling, $g_D$,
but can also lead to quantitatively different signatures once the DP is
discovered. In this setup, assuming complex scalar DM to satisfy CMB
constraints, parameter space regions where the DP decays invisibly are found to
be somewhat preferred if PM mass limits from direct searches at the LHC and our
toy model setup are all taken seriously. High precision searches for, or
measurements of, the $e^+e^- \to \gamma +{\rm DP}$ process at Belle II are
shown to provide some of the strongest future constraints on this scenario.
