  In the first Born approximation we study the reactions $K\phi\to\pi K$, $\rho
K$, $\pi K^*$, and $\rho K^*$ with quark-antiquark annihilation and creation.
Transition amplitudes are derived with the development in spherical harmonics
of the relative-motion wave functions of the two initial mesons and of the two
final mesons so that parity is conserved and the total angular momentum of the
final mesons equals the one of the initial mesons. Unpolarized cross sections
are calculated from the transition amplitudes that also contain mesonic
quark-antiquark relative-motion wave functions and transition potentials for
quark-antiquark annihilation and creation. Notable temperature dependence of
the cross sections is shown. While the cross sections for $K\phi\to\rho K$,
$K\phi\to\pi K^*$, and $K\phi\to\rho K^*$ may be of the millibarn scale, the
cross section for $K\phi\to\pi K$ is very small.
