In this work, we characterize cluster-invariant point processes for critical branching spatial processes on R d for all large enough d when the motion law is $α$-stable or has a finite discrete range. More precisely, when the motion is $α$-stable with $α$ $\le$ 2 and the offspring law $μ$ of the branching process has an heavy tail such that $μ$(k) $\sim$ k --2--$β$ , then we need the dimension d to be strictly larger than the critical dimension $α$/$β$. In particular, when the motion is Brownian and the offspring law $μ$ has a second moment, this critical dimension is 2. Contrary to the previous work of Bramson, Cox and Greven in [BCG97] whose proof used PDE techniques, our proof uses probabilistic tools only.