  Rare kaon decays are excellent probes of light, new weakly-coupled particles.
If such particles $X$ couple preferentially to muons, they can be produced in
$K\to \mu \nu X$ decays. In this letter we evaluate the future sensitivity for
this process at NA62 assuming $X$ decays either invisibly or to di-muons. Our
main physics target is the parameter space that resolves the $(g-2)_\mu$
anomaly, where $X$ is a gauged $L_\mu-L_\tau$ vector or a muon-philic scalar.
The same parameter space can also accommodate dark matter freeze out or reduce
the tension between cosmological and local measurements of $H_0$ if the new
force decays to dark matter or neutrinos, respectively. We show that for
invisible $X$ decays, a dedicated single muon trigger analysis at NA62 could
probe much of the remaining $(g-2)_\mu$ favored parameter space. Alternatively,
if $X$ decays to muons, NA62 can perform a di-muon resonance search in $K\to 3
\mu \nu$ events and greatly improve existing coverage for this process.
Independently of its sensitivity to new particles, we find that NA62 is also
sensitive to the Standard Model predicted rate for $K \to 3\mu \nu$, which has
never been measured.
