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Cross sections for the reactions $e^+ e^-\to K_S^0 K_L^0$, $K_S^0 K_L^0 π^+π^-$, $K_S^0 K_S^0 π^+π^-$, and $K_S^0 K_S^0 K^+K^-$ from events with initial-state radiation

We study the processes $e^+ e^-\to K_S^0 K_L^0 γ$, $K_S^0 K_L^0 π^+π^-γ$, $K_S^0 K_S^0 π^+π^-γ$, and $K_S^0 K_S^0 K^+K^-γ$, 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 $ϕ(1020)$ 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 $e^+ e^-\to K_S^0 K_L^0 π^+π^-$, $K_S^0 K_S^0 π^+π^-$, 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 $K_S^0 K_L^0 π^+π^-$, $K_S^0 K_S^0 π^+π^-$, and $K_S^0 K_S^0 K^+K^-$ final states.