  In the framework of the low-energy effective theory containing in addition to
the Standard Model fields heavy neutral leptons (HNLs), we compute the decay
rates of neutral and charged kaons into HNLs. We consider both
lepton-number-conserving and lepton-number-violating four-fermion operators,
taking into account also the contribution of active-heavy neutrino mixing.
Assuming that the produced HNLs are long-lived, we perform simulations and
calculate the sensitivities of future long-lived-particle (LLP) detectors at
the high-luminosity LHC as well as the near detector of the Deep Underground
Neutrino Experiment (DUNE-ND) to the considered scenario. When applicable, we
also recast the existing bounds on the minimal mixing case obtained by NA62,
T2K, and PS191. Our findings show that while the future LHC LLP detectors can
probe currently allowed parameter space only in certain benchmark scenarios,
DUNE-ND should be sensitive to parameter space beyond the current bounds in
almost all the benchmark scenarios and for some of the effective operators
considered it can even probe new-physics scales in excess of 3000 TeV.
