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Extensions of the Standard Model featuring light vector bosons have beenexplored with the goal of resolving certain tensions between theory andexperiment, among them the discrepancy in the anomalous magnetic moment of themuon, $\Delta a_{\mu}$. In particular, this is the case of a minimalconstruction including a leptophilic, strictly flavour violating, vector boson$Z^\prime$. These new vector bosons are also well-motivated dark matterportals, with non-trivial couplings to stable, weakly interacting states whichcan account for the correct dark matter density. Here we study the prospects ofa Standard Model extension (via a vector boson and a fermionic dark mattercandidate) concerning signatures at the LHC, and at future lepton and hadroncolliders.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 finalstates with missing energy (in the form of neutrinos and/or dark matter). Ourfindings suggest that a future muon collider offers the best prospects to probethis model (together with searches for dilepton pairs and missing energysignatures at the FCC-ee running at the $Z$-pole); moreover, thecomplementarity of the different future high-energy colliders is also paramountto probing distinct $Z^\prime$ mass regimes.

cLFV leptophilic $Z^\prime$ as a dark matter portal: prospects for colliders