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The discovery of lepton flavour violating interactions will be strikingevidence 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 potentialof current and future high-energy colliders to probe lepton flavour violationin the $\tau$ sector. Based on this potential we determine the expectedconstraints on parameters of new physics in the context of the Type-II SeesawModel, the Left-Right Symmetric Model, and the Minimal Supersymmetric StandardModel. The existing and ongoing 13 TeV run of the Large Hadron Collider has thepotential to produce constraints that outperform the existing $e^+ e^-$collider limits for the $\tau^\mp \to \mu^\pm \mu^\mp \mu^\mp$ decay andachieve a branching fraction limit of $\lesssim 10^{-8}$. With a futurecircular $e^+ e^-$ collider, constraints on the $\tau \to l \mu \mu$ branchingfractions could reach as low as a few times $10^{-12}$.

Prospects for new physics in $\tau \to l \mu \mu$ at current and future colliders