We study the processes $e^+ e^-to 2(pi^+pi^-)pi^0gamma$, $2(pi^+pi^-)etagamma$, $K^+ K^-pi^+pi^-pi^0gamma$ and $K^+ K^-pi^+pi^-etagamma$ with the hard photon radiated from the initial state. About 20000, 4300, 5500 and 375 fully reconstructed events, respectively, are selected from 232 fb$^{-1}$ of BaBar data. The invariant mass of the hadronic final state defines the effective $e^+ e^-$ center-of-mass energy, so that the obtained cross sections from the threshold to about 5 GeV can be compared with corresponding direct epem measurements, currently available only for the $etapi^+pi^-$ and $omegapi^+pi^-$ submodes of the $e^+ e^-to 2(pi^+pi^-)pi^0$ channel. Studying the structure of these events, we find contributions from a number of intermediate states, and we extract their cross sections where possible. In particular, we isolate the contribution from $e^+ e^-toomega(782)pi^+pi^-$ and study the $omega(1420)$ and $omega(1650)$ resonances. In the charmonium region, we observe the $J/psi$ in all these final states and several intermediate states, as well as the $psi(2S)$ in some modes, and we measure the corresponding branching fractions.
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