For a time dependent family of probability measures $(ρ_t)_{t\ge 0}$ we consider a kinetic-type evolution equation $\partial φ_t/\partial t + φ_t = \widehat{Q} φ_t$ where $\widehat{Q}$ is a smoothing transform and $φ_t$ is the Fourier--Stieltjes transform of $ρ_t$. Assuming that the initial measure $ρ_0$ belongs to the domain of attraction of a stable law, we describe asymptotic properties of $ρ_t$, 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 $(ρ_t)_{t\ge 0}$ that refines the corresponding construction proposed in Bassetti and Ladelli [Ann. Appl. Probab. 22(5): 1928--1961, 2012].