We study a one-dimensional kinetic stochastic model driven by a L{é}vy process with a non-linear time-inhomogeneous drift. More precisely, the process $(V,X)$ is considered, where $X$ is the position of the particle and its velocity $V$ is the solution of a stochastic differential equation with a drift of the form $t^{-β}F(v)$. The driving process can be a stable L{é}vy process of index $α$ or a general L{é}vy process under appropriate assumptions. The function $F$ satisfies a homogeneity condition and $β$ is non-negative. The behavior in large time of the process $(V,X)$ is proved and the precise rate of convergence is pointed out by using stochastic analysis tools. To this end, we compute the moment estimates of the velocity process.