For a time dependent family of probability measure $(\mu_t)_{t\he 0}$ we consider a kinetic-type evolution equation $\partial \mu_t/\partial t + \mu_t = Q \mu_t$, where $Q$ is the smoothing transformation. During the talk we will present probabilistic representation of a solution of this equation in terms of continuous time branching random walks. Moreover, assuming that $\mu_0$ belongs to the domain of attraction of a stable law, we describe asymptotic behaviour of $\mu_t$. Literature: [1] Bogus, B., Marynych, SPA 2020 [2] B., Kolesko, Meiners, EJP 2021 [3] B., Dyszewski, Marynych, SPA 2023