  We study the asymptotic behavior as $t \to \infty$ of a time-dependent family
$(\mu_t)_{t \geq 0}$ of probability measures on $\mathbb{R}$ solving the
kinetic-type evolution equation $\partial_t \mu_t + \mu_t = Q(\mu_t)$ where $Q$
is a smoothing transformation on $\mathbb{R}$. This problem has been
investigated earlier, e.g. by Bassetti and Ladelli [Ann. Appl. Probab. 22(5):
1928-1961, 2012] and Bogus, Buraczewski and Marynych [Stochastic Process. Appl.
130(2):677-693, 2020]. Combining the refined analysis of the latter paper,
which provides a probabilistic description of the solution $\mu_t$ as the law
of a suitable random sum related to a continuous-time branching random walk at
time $t$, with recent advances in the analysis of the extremal positions in the
branching random walk we are able to solve the remaining case that has been
left open until now. In the course of our work, we significantly weaken the
assumptions in the literature that guarantee the existence (and uniqueness) of
a solution to the evolution equation $\partial_t \mu_t + \mu_t = Q(\mu_t)$.
