  We study two kinds of inelastic meson-meson scattering. The first kind is
inelastic 2-to-2 meson-meson scattering that is governed by quark interchange
as well as quark-antiquark annihilation and creation. Cross-section formulas
are provided to get unpolarized cross sections for $\pi K \to \rho K^\ast$ for
$I=1/2$, $\pi K^\ast \to \rho K$ for $I=1/2$, $\pi K^\ast \to \rho K^\ast$ for
$I=1/2$, and $\rho K \to \rho K^\ast$ for $I=1/2$. Near threshold, quark
interchange dominates the reactions near the critical temperature. The second
kind is 2-to-1 meson-meson scattering with the process that a quark in an
initial meson and an antiquark in another initial meson annihilate into a gluon
and subsequently the gluon is absorbed by the other antiquark or quark. The
transition potential for the process is derived. Four Feynman diagrams at tree
level contribute to the 2-to-1 meson-meson scattering. Starting from the
$S$-matrix element, the isospin-averaged unpolarized cross section with
transition amplitudes is derived. The cross sections for $\pi \pi \to \rho$ and
$\pi K \to K^*$ decrease with increasing temperature.
