  The processes $e^+e^-\to K^0_{\scriptscriptstyle S}K^\pm\pi^{\mp}\pi^0$ and
$e^+e^-\to K^0_{\scriptscriptstyle S}K^\pm\pi^{\mp}\eta$ 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
$K^{*0}(K\pi)^0$, $K^{*}(892)^\pm(K\pi)^\mp$ and $K^0_{\scriptscriptstyle
S}K^\pm\rho^{\mp}$ are studied. The $J/\psi$ is observed in all of these
channels, and corresponding branching fractions are measured.
