, where the photon is radiated from the initial state. About 84000, 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)ππγ reaction, and confirm the presence of the Y (2175) resonance in the φ(1020)f0(980) and K + K − f0(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.