  We study the processes $e^+ e^-\to K_S^0 K_L^0 \gamma$, $K_S^0 K_L^0
\pi^+\pi^-\gamma$, $K_S^0 K_S^0 \pi^+\pi^-\gamma$, and $K_S^0 K_S^0
K^+K^-\gamma$, 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 $K_S^0 K_L^0$ 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^-\to K_S^0 K_L^0 $ 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 $e^+ e^-\to K_S^0 K_L^0 \pi^+\pi^-$,
$K_S^0 K_S^0 \pi^+\pi^-$, and $K_S^0 K_S^0 K^+K^-$ cross sections, and study
the intermediate resonance structures. We obtain the first observations of
\jpsi decay to the $K_S^0 K_L^0 \pi^+\pi^-$, $K_S^0 K_S^0 \pi^+\pi^-$, and
$K_S^0 K_S^0 K^+K^-$ final states.
