We estimate that the LHC could set bounds $BR(Z\to \mu^\pm e^\mp) < 4.1\times 10^{-7}$ and $BR(Z \to \tau^\pm \mu^\mp)< 3.5\times 10^{-6}$ (at 95% C.L.) with 20 fb$^{-1}$ of data at 8 TeV. A similar sensitivity can be anticipated for $Z\to \tau^\pm e^\mp$, because we consider leptonic $\tau$ decays such that $Z\to \tau^+ \mu^- \to e^+ \mu^- +$ invisible. These limits can be compared to the LEP1 bounds of order $10^{-5} \to 10^{-6}$. Such collider searches are sensitive to a flavour-changing effective $Z$ coupling which is energy dependent, so are complementary to bounds obtained from $\tau \to 3 \mu$ and $\mu \to 3e$.