We study the processes e+e- -> K0S K0L gamma, K0S K0L pi+ pi- gamma, K0S K0S pi+ pi- gamma, and K0S K0S K+ K- gamma, 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 469 fb-1 of data collected with the BABAR detector at SLAC. We observe the phi(1020) resonance in the K0S K0L 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- -> K0S K0L 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- -> K0S K0L pi+ pi-, K0S K0S pi+ pi-, and K0S K0S K+ K- cross sections and study the intermediate resonance structures. We obtain the first observations of J/Psi decay to the K0S K0L pi+ pi-, K0S K0S pi+ pi-, and K0S K0S K+ K- final states.