We study the processes e+e-→KS0KL0γ, KS0KL0π+π-γ, KS0KS0π+π-γ, and KS0KS0K+K-γ, where the photon is radiated from the initial state, providing cross section measurements for the hadronic states over a continuum of center-of-mass energies. The results are based on 469fb-1 of data collected with the BABAR detector at SLAC. We observe the φ(1020) resonance in the KS0KL0 final state and measure the product of its electronic width and branching fraction with about 3% uncertainty. We present a measurement of the e+e-→KS0KL0 cross section in the energy range from 1.06 to 2.2 GeV and observe the production of a resonance at 1.67 GeV. We present the first measurements of the e+e-→KS0KL0π+π-, KS0KS0π+π-, and KS0KS0K+K- cross sections and study the intermediate resonance structures. We obtain the first observations of J/ψ decay to the KS0KL0π+π-, KS0KS0π+π-, and KS0KS0K+K- final states. © 2014 American Physical Society.