  We study a one-dimensional kinetic stochastic model driven by a L{\'e}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 $t^{-\beta}F(v)$. The driving process can be a stable L{\'e}vy
process of index $\alpha$ or a general L{\'e}vy process under appropriate
assumptions. The function $F$ satisfies a homogeneity condition and $\beta$ 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.
