We study a class of kinetic-type differential equations $\partial φ_t/\partial t+φ_t=\widehat{\mathcal{Q}}φ_t$, where $\widehat{\mathcal{Q}}$ is an inhomogeneous smoothing transform and, for every $t\geq 0$, $φ_t$ is the Fourier--Stieltjes transform of a probability measure. We show that under mild assumptions on $\widehat{\mathcal{Q}}$ the above differential equation possesses a unique solution and represent this solution as the characteristic function of a certain stochastic process associated with the continuous time branching random walk pertaining to $\widehat{\mathcal{Q}}$. Establishing limit theorems for this process allows us to describe asymptotic properties of the solution, as $t\to\infty$.