The processes $e^+e^-to K^0_{scriptscriptstyle S}K^pmpi^{mp}pi^0$ and $e^+e^-to K^0_{scriptscriptstyle S}K^pmpi^{mp}eta$ are studied over a continuum of energies from threshold to 4 GeV with the initial-state photon radiation method. Using 454 fb$^{-1}$ 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 $K^{*0}(Kpi)^0$, $K^{*}(892)^pm(Kpi)^mp$ and $K^0_{scriptscriptstyle S}K^pmrho^{mp}$ are studied. The $J/psi$ is observed in all of these channels, and corresponding branching fractions are measured.