© 2017 American Physical Society. The processes e+e-→KS0K±π π0 and e+e-→KS0K±π η are studied over a continuum of energies from threshold to 4 GeV with the initial-state photon radiation method. Using 454 fb-1 of data collected with the BABAR detector at the SLAC PEP-II storage ring, the first measurements of the cross sections for these processes are obtained. The intermediate resonance structures from K∗0(Kπ)0, K∗(892)±(Kπ) , and KS0K±ρ are studied. The J/ψ is observed in all of these channels, and corresponding branching fractions are measured.