  We provide general effective-theory arguments relating present-day
discrepancies in semi-leptonic $B$-meson decays to signals in kaon physics, in
particular lepton-flavour violating ones of the kind $K \to (\pi) e^\pm
\mu^\mp$. We show that $K$-decay branching ratios of around $10^{-12} -
10^{-13}$ are possible, for effective-theory cutoffs around $5-15$ TeV
compatible with discrepancies in $B\to K^{(\ast)} \mu\mu$ decays. We perform a
feasibility study of the reach for such decays at LHCb, taking $K^+ \to \pi^+
\mu^\pm e^\mp$ as a benchmark. In spite of the long lifetime of the $K^+$
compared to the detector size, the huge statistics anticipated as well as the
overall detector performance translate into encouraging results. These include
the possibility to reach the $10^{-12}$ ballpark, and thereby significantly
improve current limits. Our results advocate LHC's high-luminosity Upgrade
phase, and support analogous sensitivity studies at other facilities. Given the
performance uncertainties inherent in the Upgrade phase, our conclusions are
based on a range of assumptions we deem realistic on the particle
identification performance as well as on the kinematic reconstruction
thresholds for the signal candidates.
