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→μνX decays. We evaluate the future sensitivity for this process at NA62 assuming X decays either invisibly or to dimuons. Our main physics target is the parameter space that resolves the (g-2)_{μ} anomaly, where X is a gauged L_{μ}-L_{τ} vector or a muonphilic 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)_{μ} favored parameter space. Alternatively, if X decays to muons, NA62 can perform a dimuon resonance search in K→3μν 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→3μν, which has never been measured.