  The discovery of lepton flavour violating interactions will be striking
evidence for physics beyond the Standard Model. Focusing on the three decays
$\tau^\mp \to \mu^\pm \mu^\mp \mu^\mp$, $\tau^\mp \to e^\pm \mu^\mp \mu^\mp$
and $\tau^\mp \to e^\mp \mu^\mp \mu^\pm$, we evaluate the discovery potential
of current and future high-energy colliders to probe lepton flavour violation
in the $\tau$ sector. Based on this potential we determine the expected
constraints on parameters of new physics in the context of the Type-II Seesaw
Model, the Left-Right Symmetric Model, and the Minimal Supersymmetric Standard
Model. The existing and ongoing 13 TeV run of the Large Hadron Collider has the
potential to produce constraints that outperform the existing $e^+ e^-$
collider limits for the $\tau^\mp \to \mu^\pm \mu^\mp \mu^\mp$ decay and
achieve a branching fraction limit of $\lesssim 10^{-8}$. With a future
circular $e^+ e^-$ collider, constraints on the $\tau \to l \mu \mu$ branching
fractions could reach as low as a few times $10^{-12}$.
