  We explore the feasibility of using TeV-energy muons to probe
lepton-flavor-violating (LFV) processes mediated by an axion-like particle
(ALP) $a$ with mass $\mathcal{O}(10~\textrm{GeV})$. We focus on $\mu\tau$ LFV
interactions and assume that the ALP is coupled to a dark state $\chi$, which
can be either less or more massive than $a$. Such a setup is demonstrated to be
consistent with $\chi$ being a candidate for dark matter, in the experimentally
relevant regime of parameters. We consider the currently operating NA64-$\mu$
experiment and proposed FASER$\nu$2 detector as both the target and the
detector for the process $\mu A \to \tau A\, a$, where $A$ is the target
nucleus. We also show that a possible future active muon fixed-target
experiment operating at a 3 TeV muon collider or in its preparatory phase can
provide an impressive reach for the LFV process considered, with future
FASER$\nu$2 data providing a pilot study towards that goal. The implications of
the muon anomalous magnetic moment $(g-2)_\mu$ measurements for the underlying
model, in case of a positive signal, are also examined, and a sample UV
completion is outlined.
