  Scaling limits for continuous-time branching processes with discrete state
space are provided as the initial state tends to infinity. Depending on the
finiteness or non-finiteness of the mean and/or the variance of the offspring
distribution, the limits are in general time-inhomogeneous Gaussian processes,
time-inhomogeneous generalized Ornstein-Uhlenbeck type processes or
continuous-state branching processes. We also provide transfer results showing
how specific asymptotic relations for the probability generating function of
the offspring distribution carry over to those of the one-dimensional
distributions of the branching process.
