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We study the processes $e^+ e^-\to K^+ K^- \pi^+\pi^-\gamma$,$K^+K^-\pi^0\pi^0\gamma$ and $K^+ K^- K^+ K^-\gamma$, where the photon isradiated from the initial state. About 34600, 4400 and 2300 fully reconstructedevents, respectively, are selected from 232 \invfb of \babar data. Theinvariant mass of the hadronic final state defines the effective \epemcenter-of-mass energy, so that the $K^+ K^- \pi^+\pi^-\gamma$ data can becompared with direct measurements of the $e^+ e^-\to K^+K^- \pipi$ reaction; nodirect measurements exist for the $e^+ e^-\to K^+ K^- \pi^0\pi^0$ or $\epem\toK^+ K^- K^+ K^-$ reactions. Studying the structure of these events, we findcontributions from a number of intermediate states, and we extract their crosssections where possible. In particular, we isolate the contribution from $e^+e^-\to\phi(1020) f_{0}(980)$ and study its structure near threshold. In thecharmonium region, we observe the $J/\psi$ in all three final states andseveral intermediate states, as well as the $\psi(2S)$ in some modes, andmeasure the corresponding branching fractions. We see no signal for the Y(4260)and obtain an upper limit of$\BR_{Y(4260)\to\phi\pi^+\pi^-}\cdot\Gamma^{Y}_{ee}<0.4 \ev$ at 90% C.L.

The $e^+ e^-\to K^+ K^- \pi^+\pi^-$, $K^+ K^- \pi^0\pi^0$ and $K^+ K^- K^+ K^-$ Cross Sections Measured with Initial-State Radiation