We study the processes e +e -→K +K -π +π -γ, K +K -π0π0γ, and K +K -K +K -γ, where the photon is radiated from the initial state. About 84000, 8000, and 4200 fully reconstructed events, respectively, are selected from 454fb -1 of BABAR data. The invariant mass of the hadronic final state defines the e +e - center-of-mass energy, so that the K +K -π +π -γ data can be compared with direct measurements of the e +e -→K +K -π +π - reaction. No direct measurements exist for the e +e -→K +K -π0π0 or e +e -→K +K -K +K - reactions, and we present an update of our previous result based on a data sample that is twice as large. Studying the structure of these events, we find contributions from a number of intermediate states and extract their cross sections. In particular, we perform a more detailed study of the e +e -→φ(1020)ππγ reaction and confirm the presence of the Y(2175) resonance in the φ(1020)f 0(980) and K +K -f 0(980) modes. In the charmonium region, we observe the J/ψ in all three final states and in several intermediate states, as well as the ψ(2S) in some modes, and measure the corresponding products of branching fraction and electron width. © 2012 American Physical Society.