  A $p$-jump process is a piecewise deterministic Markov process with jumps by
a factor of $p$. We prove a limit theorem for such processes on the unit
interval. Via duality with respect to probability generating functions, we
deduce limiting results for the survival probabilities of time-homogeneous
branching processes with arbitrary offspring distributions, underlying binomial
disasters. Extending this method, we obtain corresponding results for
time-inhomogeneous birth-death processes underlying time-dependent binomial
disasters and continuous state branching processes with $p$-jumps.
