We study the processes e + e − → K + K − π + π − γ , K + K − π 0 π 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 0 π 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 ) π π γ 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 ( 2 S ) in some modes, and measure the corresponding products of branching fraction and electron width.