  For a time dependent family of probability measures $(\rho_t)_{t\ge 0}$ we
consider a kinetic-type evolution equation $\partial \phi_t/\partial t + \phi_t
= \widehat{Q} \phi_t$ where $\widehat{Q}$ is a smoothing transform and $\phi_t$
is the Fourier--Stieltjes transform of $\rho_t$. Assuming that the initial
measure $\rho_0$ belongs to the domain of attraction of a stable law, we
describe asymptotic properties of $\rho_t$, as $t\to\infty$. We consider the
critical regime when the standard normalization leads to a degenerate limit and
find an appropriate scaling ensuring a non-degenerate self-similar limit. Our
approach is based on a probabilistic representation of probability measures
$(\rho_t)_{t\ge 0}$ that refines the corresponding construction proposed in
Bassetti and Ladelli [Ann. Appl. Probab. 22(5): 1928--1961, 2012].
