  This work deals with a class of one-dimensional measure-valued kinetic
equations, which constitute extensions of the Kac caricature. It is known that
if the initial datum belongs to the domain of normal attraction of an
\alpha-stable law, the solution of the equation converges weakly to a suitable
scale mixture of centered \alpha-stable laws. In this paper we present explicit
exponential rates for the convergence to equilibrium in Kantorovich-Wasserstein
distances of order p>\alpha, under the natural assumption that the distance
between the initial datum and the limit distribution is finite. For \alpha=2
this assumption reduces to the finiteness of the absolute moment of order p of
the initial datum. On the contrary, when \alpha<2, the situation is more
problematic due to the fact that both the limit distribution and the initial
datum have infinite absolute moment of any order p >\alpha. For this case, we
provide sufficient conditions for the finiteness of the Kantorovich-Wasserstein
distance.
