  Consider compound Poisson processes with negative drift and no negative
jumps, which converge to some spectrally positive L\'evy process with non-zero
L\'evy measure. In this paper we study the asymptotic behavior of the local
time process, in the spatial variable, of these processes killed at two
different random times: either at the time of the first visit of the L\'evy
process to 0, in which case we prove results at the excursion level under
suitable conditionings; or at the time when the local time at 0 exceeds some
fixed level. We prove that finite-dimensional distributions converge under
general assumptions, even if the limiting process is not c\`adl\`ag. Making an
assumption on the distribution of the jumps of the compound Poisson processes,
we strengthen this to get weak convergence. Our assumption allows for the
limiting process to be a stable L\'evy process with drift.
  These results have implications on branching processes and in queueing
theory, namely, on the scaling limit of binary, homogeneous Crump-Mode-Jagers
processes and on the scaling limit of the Processor-Sharing queue length
process.
