  We consider inhomogeneous branching diffusions on an infinite domain of
$\mathbb{R}^d$. The first aim of this article is to derive a general criterium
under which the size process (number of particles) and the genealogy of the
particle system become undistinguishable from the ones of an $\alpha$-stable
CSBP, with $\alpha\in(1,2)$. The branching diffusion is encoded as a random
metric space capturing all the information about the positions and the
genealogical structure of the population. Our convergence criterium is based on
the convergence of the moments for random metric spaces, which in turn can be
efficiently computed through many-to-few formulas. It requires an extension of
the method of moments to general CSBPs (with or without finite second moment).
  In a recent work, Tourniaire introduced a branching Brownian motion which can
be thought of as a toy model for fluctuating pushed fronts. The size process
was shown to converge to an $\alpha$-stable CSBP and it was conjectured that a
genealogical convergence should occur jointly. We prove this result as an
application of our general methodology.
