  Pseudo-scalar particles, often referred to as axion-like-particles (ALPs),
arise in a variety of theoretical contexts. The phenomenology of such states is
typically studied assuming flavor-conserving interactions, yet they can in
principle have flavor-violating (FV) couplings to fermions. We consider this
general possibility, focusing on models where the ALP has non-negligible
coupling to the Standard Model Higgs boson $h$. For a lepton FV ALP $a$ of mass
$m_a \gtrsim 2$ GeV, $a\to \tau \ell$, where $\ell\neq \tau$ is a charged
lepton, could have $\mathcal{O}(1)$ branching fraction, leading to potentially
detectable $h \to a a \to \tau \ell \tau \ell$ at the LHC and its future
program. We examine this possibility, in light of existing bounds on FV
processes, in a general effective theory. We obtain constraints on the
effective couplings from both prompt and long-lifetime searches at the LHC;
some projections for envisioned measurements are also provided. The
implications of the recently announced first results of the muon $g-2$
measurement at Fermilab for the ALP interactions considered in our work are
also briefly discussed.
