We study the processes e(+)e(-)-> K+K-pi(+)pi(-)gamma, K+K-pi(0)pi(0)gamma and K+K-K+K-gamma, where the photon is radiated from the initial state. About 34 600, 4400 and 2300 fully reconstructed events, respectively, are selected from 232 fb(-1) of BABAR data. The invariant mass of the hadronic final state defines the effective e(+)e(-) center-of-mass energy, so that the K+K-pi(+)pi(-)gamma 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-pi(0)pi(0) or e(+)e(-)-> K+K-K+K- reactions. Studying the structure of these events, we find contributions from a number of intermediate states, and we extract their cross sections where possible. In particular, we isolate the contribution from e(+)e(-)->phi(1020)f(0)(980) and study its structure near threshold. In the charmonium region, we observe the J/psi in all three final states and several intermediate states, as well as the psi(2S) in some modes, and measure the corresponding branching fractions. We see no signal for the Y(4260) and obtain an upper limit of B-Y(4260)->phi pi(+)pi(-)center dot Gamma(Y)(ee)< 0.4 eV at 90% C.L.
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