© 2017 American Physical Society. We study the processes e+e-→KS0KL0π0γ, KS0KL0ηγ, and KS0KL0π0π0γ, where the photon is radiated from the initial state, providing cross section measurements for the hadronic final states over a continuum of center-of-mass energies. The results are based on 469 fb-1 of data collected at or near the ϒ(4S) resonance with the BABAR detector at SLAC. We present the first measurements of the e+e-→KS0KL0π0, KS0KL0η, and KS0KL0π0π0 cross sections up to a center-of-mass energy of 4 GeV and study their intermediate resonance structures. We observe J/ψ decays to all of these final states for the first time, present measurements of their J/ψ branching fractions, and search for ψ(2S) decays.