  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 is
radiated from the initial state. About 34600, 4400 and 2300 fully reconstructed
events, respectively, are selected from 232 \invfb of \babar data. The
invariant mass of the hadronic final state defines the effective \epem
center-of-mass energy, so that the $K^+ K^- \pi^+\pi^-\gamma$ data can be
compared with direct measurements of the $e^+ e^-\to K^+K^- \pipi$ reaction; no
direct measurements exist for the $e^+ e^-\to K^+ K^- \pi^0\pi^0$ or $\epem\to
K^+ K^- K^+ K^-$ 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 $e^+
e^-\to\phi(1020) f_{0}(980)$ 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
$\BR_{Y(4260)\to\phi\pi^+\pi^-}\cdot\Gamma^{Y}_{ee}<0.4 \ev$ at 90% C.L.
