  Extensions of the Standard Model featuring light vector bosons have been
explored with the goal of resolving certain tensions between theory and
experiment, among them the discrepancy in the anomalous magnetic moment of the
muon, $\Delta a_{\mu}$. In particular, this is the case of a minimal
construction including a leptophilic, strictly flavour violating, vector boson
$Z^\prime$. These new vector bosons are also well-motivated dark matter
portals, with non-trivial couplings to stable, weakly interacting states which
can account for the correct dark matter density. Here we study the prospects of
a Standard Model extension (via a vector boson and a fermionic dark matter
candidate) concerning signatures at the LHC, and at future lepton and hadron
colliders.We discuss the cross-sections of several processes leading to same-
and opposite-sign muon-tau lepton pairs in the final state, as well as final
states with missing energy (in the form of neutrinos and/or dark matter). Our
findings suggest that a future muon collider offers the best prospects to probe
this model (together with searches for dilepton pairs and missing energy
signatures at the FCC-ee running at the $Z$-pole); moreover, the
complementarity of the different future high-energy colliders is also paramount
to probing distinct $Z^\prime$ mass regimes.
