  We consider the $\phi^4$ model in one space dimension with propagation speeds
that are small deviations from a constant function. In the constant-speed case,
a stationary solution called the kink is known explicitly, and the recent work
of Kowalczyk, Martel, and Mu\~noz established the asymptotic stability of the
kink with respect to odd perturbations in the natural energy space. We show
that a stationary kink solution exists also for our class of non-constant
propagation speeds, and extend the asymptotic stability result by taking a
perturbative approach to the method of Kowalczyk, Martel, and Mu\~noz. This
requires an understanding of the spectrum of the linearization around the
variable-speed kink.
