  Branching processes and Fleming-Viot processes are two main models in
stochastic population theory. Incorporating an immigration in both models, we
generalize the results of Shiga (1990) and Birkner et al. (2005) which
respectively connect the Feller diffusion with the classical Fleming-Viot
process and the alpha-stable continuous state branching process with the
Beta(2-alpha, alpha)-generalized Fleming-Viot process. In a recent work, a new
class of probability-measure valued processes, called M-generalized
Fleming-Viot processes with immigration, has been set up in duality with the
so-called M-coalescents. The purpose of this article is to investigate the
links between this new class of processes and the continuous-state branching
processes with immigration. In the specific case of the $\alpha$-stable
branching process conditioned to be never extinct, we get that its genealogy is
given, up to a random time change, by a Beta(2-alpha, alpha-1)-coalescent.
