We implement a systematic effective field theory approach to the benchmark process $\mu\to e \gamma$, performing automated one-loop computations including dimension 6 operators and studying their anomalous dimensions. We obtain limits on Wilson coefficients of a relevant subset of lepton-flavour violating operators that contribute to the branching ratio $\mu\to e \gamma$ at one-loop. In addition, we illustrate a method to extract further constraints induced by the mixing of operators under renormalisation-group evolution. This results in limits on the corresponding Wilson coefficients directly at the high scale. The procedure can be applied to other processes as well and, as an example, we consider also lepton-flavour violating decays of the $\tau$.