  We consider a sequence of Poisson cluster point processes on $\mathbb{R}^d$:
at step $n\in\mathbb{N}_0$ of the construction, the cluster centers have
intensity $c/(n+1)$ for some $c>0$, and each cluster consists of the particles
of a branching random walk up to generation $n$ generated by a point process
with mean 1. We show that this 'critical cluster cascade' converges weakly, and
that either the limit point process equals the void process (extinction), or it
has the same intensity $c$ as the critical cluster cascade (persistence). We
obtain persistence, if and only if the Palm version of the outgrown critical
branching random walk is locally a.s. finite. This result allows us to give
numerous examples for persistent critical cluster cascades.
