We study the processes e þ e − → π þ π − π 0 π 0 π 0 γ and π þ π − π 0 π 0 ηγ 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 þ e − center-of-mass energy. From the mass spectra, the first precise measurement of the e þ e − → π þ π − π 0 π 0 π 0 cross section and the first measurement ever of the e þ e − → π þ π − π 0 π 0 η 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 ωπ 0 π 0 , ηπ þ π − , and other intermediate states are presented. We observe the J=ψ and ψð2SÞ in most of these final states and measure the corresponding branching fractions, many of them for the first time.