We study the processes e+e-→π+π-π0π0π0γ and π+π-π0π0ηγ in which an energetic photon is radiated from the initial state. The data are collected with the BABAR detector at SLAC. About 14 000 and 4700 events, respectively, are selected from a data sample corresponding to an integrated luminosity of 469 fb-1. The invariant mass of the hadronic final state defines the effective e+e- center-of-mass energy. From the mass spectra, the first precise measurement of the e+e-→π+π-π0π0π0 cross section and the first measurement ever of the e+e-→π+π-π0π0η cross section are performed. The center-of-mass energies range from threshold to 4.35 GeV. The systematic uncertainty is typically between 10% and 13%. The contributions from ωπ0π0, ηπ+π-, and other intermediate states are presented. We observe the J/ψ and ψ(2S) in most of these final states and measure the corresponding branching fractions, many of them for the first time.