We study the processes e+e- --> K+ K- pi+pi-gamma, K+ K- pi0pi0gamma, and K+
K- K+ K-gamma, 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 \epem center-of-mass energy, so that the K+ K- pi+pi- data can be
compared with direct measurements of the e+e- --> K+ K- pi+pi- reaction. No
direct measurements exist for the e+e- --> K+ K-pi0pi0 or e+e- --> K+ K-K+ K-
reactions, and we present an update of our previous result with doubled
statistics. 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- --> phi(1020)pipigamma
reaction, and confirm the presence of the Y(2175) resonance in the phi(1020)
f0(980) and K+K-f0(980) modes. In the charmonium region, we observe the J/psi
in all three final states and in several intermediate states, as well as the
psi(2S) in some modes, and measure the corresponding product of branching
fraction and electron width.Comment: 35 pages, 42 figure