We study the processes e^+e^- → K^+K^-π^+π^-γ, K^+K^-π^0π^0 γ, and K^+K^-K^+K^- γ, where the photon is radiated from the initial state. About 84 000, 8000, and 4200 fully reconstructed events, respectively, are selected from 454 fb^(-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^-π^0π^0 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
