  In leptonic flavour models with discrete flavour symmetries, couplings
between flavons and leptons can result in special flavour structures after they
gain vacuum expectation values. At the same time, they can also contribute to
the other lepton-flavour-violating processes. We study the flavon-induced LFV
3-body charged lepton decays and radiative decays and we take as example the
$A_4$ discrete symmetry. In $A_4$ models, a $Z_3$ residual symmetry roughly
holds in the charged lepton sector for the realisation of tri-bimaximal mixing
at leading order. The only processes allowed by this symmetry are $\tau^-\to
\mu^+ e^- e^-, e^+ \mu^- \mu^-$, and the other 3-body and all radiative decays
are suppressed by small $Z_3$-breaking effects. These processes also depend on
the representation the flavon is in, whether pseudo-real (case i) or complex
(case ii). We calculate the decay rates for all processes for each case and
derive their strong connection with lepton flavour mixing. In case i, sum rules
for the branching ratios of these processes are obtained, with typical examples
$\text{Br}(\tau^-\to \mu^+ e^- e^-) \approx \text{Br}(\tau^-\to e^+ \mu^-
\mu^-)$ and $\text{Br}(\tau^-\to e^- \gamma) \approx \text{Br}(\tau^-\to \mu^-
\gamma)$. In case ii, we observe that the mixing between two $Z_3$-covariant
flavons plays an important role. All processes are suppressed by charged lepton
masses and current experimental constraints allow the electroweak scale and the
flavon masses to be around hundreds of GeV. Our discussion can be generalised
in other flavour models with different flavour symmetries.
