  We study the processes $e^+ e^-\to 2(\pi^+\pi^-)\pi^0\gamma$,
$2(\pi^+\pi^-)\eta\gamma$, $K^+ K^-\pi^+\pi^-\pi^0\gamma$ and $K^+
K^-\pi^+\pi^-\eta\gamma$ 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
$\eta\pi^+\pi^-$ and $\omega\pi^+\pi^-$ 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 $e^+
e^-\to\omega(782)\pi^+\pi^-$ 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.
