  We revisit charged lepton flavour in-flight conversions, in which a beam of
electrons or muons is directed onto a fixed target, $e + N \to \mu +N$, $e + N
\to \tau +N$ and $\mu + N \to \tau +N$, focusing on elastic interactions with a
nucleus $N$. After a general discussion of this observable, we carry a full
phenomenological analysis in the framework of minimal Standard Model extensions
via sterile neutrinos, with a strong emphasis on the r\^ole of the increasingly
more stringent constraints arising from other (low-energy) charged lepton
flavour violation observables. Despite the potential interest of this
observable, in particular in the light of certain upcoming facilities with the
capability of very intense lepton beams, our study suggests that due to current
bounds on three-body decays ($\ell_i \to 3 \ell_j$) and $\mu-e$ conversion in
Nuclei, the expected number of conversions in such a minimal framework is
dramatically reduced. An experimental observation of such a conversion would
thus signal the presence of another source of flavour violation, possibly at
tree-level.
