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 $Bto K^{(ast)} mumu$ 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 LHCs 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.
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