We study the processes e + e − → 2(π + π − )π 0 γ, 2(π + π − )ηγ, K + K − π + π − π 0 γ and K + K − π + π − ηγ 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 e + e − measurements, currently available only for the ηπ + π − and ωπ + π − submodes of the e + e − → 2(π + π − )π 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 − → ω(782)π + π − and study the ω(1420) and ω (1650) resonances. In the charmonium region, we observe the J/ψ in all these final states and several intermediate states, as well as the ψ(2S) in some modes, and we measure the corresponding branching fractions.