Cross sections for the reactions e^+ e^-to K⁰_S K⁰_Lπ⁰, K⁰_S K⁰_Lη, and K⁰_S K⁰_Lπ⁰π⁰ from events with initial-state radiation

We study the processes $e^+ e^-\to K^0_S K^0_L\pi^0\gamma$, $K^0_S K^0_L\eta\gamma$, and $K^0_S K^0_L\pi^0\pi^0\gamma$, where the photon is radiated from the initial state, providing cross section measurements for the hadronic final states over a continuum of center-of-mass energies. The results are based on 469~$fb^{-1}$ of data collected at or near the $\Upsilon(4S)$~ resonance with the BaBar detector at SLAC. We present the first measurements of the $e^+ e^-\to K^0_S K^0_L\pi^0$, $K^0_S K^0_L\eta$, and $K^0_S K^0_L\pi^0\pi^0$ cross sections up to a center-of-mass energy of 4~GeV, and study their intermediate resonance structures. We observe $J/\psi$ decays to all of these final states for the first time, present measurements of their $J/\psi$ branching fractions, and search for $\psi(2S)$ decays.