  We propose a class of non-Markov population models with continuous or
discrete state space via a limiting procedure involving sequences of rescaled
and randomly time-changed Galton--Watson processes. The class includes as
specific cases the classical continuous-state branching processes and Markov
branching processes. Several results such as the expressions of moments and the
branching inequality governing the evolution of the process are presented and
commented. The generalized Feller branching diffusion and the fractional Yule
process are analyzed in detail as special cases of the general model.
