On the distribution of the first jump over a high barrier for a generalized Poisson process with drift

Let $\xi(t)$ be a process with independent increments and finite number of jumps. Define $\eta_x=\inf\{t\colon\xi(t)\ge x\}$ and $\chi_x=\xi(\eta_x)-x(x>0)$.
For the limit distribution
$$\lim_{x\to\infty}\mathbf P\{\chi_x\le x\}$$
explicit expresions are given.