Cross sections for the reactions $e^{+}{e}^{-}\to K_S^0{K}_L^0{\pi }^0$, $K_S^0{K}_L^0\eta$, and $K_S^0{K}_L^0{\pi }^0{\pi }^0$ from events with initial-state radiation

We study the processes $e^{+}{e}^{-}\to K_S^0{K}_L^0{\pi }^0\gamma$, $K^0_S K^0_L\eta\gamma$,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~$f{b}^{-1}$ of data collected at or near the $\mathrm{{\rm Y}}(4S)$~ resonance with the BaBar detector at SLAC. We present the first measurements of the $e^{+}{e}^{-}\to K_S^0{K}_L^0{\pi }^0$, $K_S^0{K}_L^0\eta$, 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 $J\mathrm{/}\psi$ decays to all of these final states for the first time, present measurements of their $J\mathrm{/}\psi$ branching fractions, and search for $\psi (2S)$ decays.