  Rare kaon decays with missing energy, $K\to\pi$+$E_{\rm miss}$, have received
considerable attention because their rates can be calculated quite precisely
within the standard model (SM), where the missing energy is carried away by an
undetected neutrino-antineutrino pair. Beyond the SM, clean theoretical
predictions can also be made regarding these processes. One such prediction is
the so-called Grossman-Nir (GN) bound, which states that the branching
fractions of the $K_L$ and $K^+$ modes must satisfy the relation
$\mathcal{B}(K_L\to\pi^0$+$E_{\rm
miss})\lesssim4.3\,\mathcal{B}(K^+\to\pi^+$+$E_{\rm miss})$ and applies within
and beyond the SM, as long as the hadronic transitions change isospin by
$\Delta I=1/2$. In this paper we extend the study of these modes to include
new-physics scenarios where the missing energy is due to unobserved
lepton-number-violating neutrino pairs, invisible light new scalars, or pairs
of such scalars. The new interactions are assumed to arise above the
electroweak scale and described by an effective field theory. We explore the
possibility of violating the GN bound through $\Delta I=3/2$ contributions to
the $K\to\pi$ transitions within these scenarios and find that large violations
are only possible in the case where the missing energy is due to an invisible
light new scalar.
