  We consider the setting of either a general non-local branching particle
process or a general non-local superprocess, in both cases, with and without
immigration. Under the assumption that the mean semigroup has a
Perron-Frobenious type behaviour for the immigrated mass, as well as the
existence of second moments, we consider necessary and sufficient conditions
that ensure limiting distributional stability. More precisely, our first main
contribution pertains to proving the asymptotic Kolmogorov survival probability
and Yaglom limit for critical non-local branching particle systems and
superprocesses under a second moment assumption on the offspring distribution.
Our results improve on existing literature by removing the requirement of
bounded offspring in the particle setting [21] and generalising [43] to allow
for non-local branching mechanisms. Our second main contribution pertains to
the stability of both critical and sub-critical non-local branching particle
systems and superprocesses with immigration. At criticality, we show that the
scaled process converges to a Gamma distribution under a necessary and
sufficient integral test. At subcriticality we show stability of the process,
also subject to an integral test. In these cases, our results complement
classical results for (continuous-time) Galton-Watson processes with
immigration and continuous-state branching processes with immigration; see
[22,40,42,48,51], among others. In the setting of superprocesses, the only work
we know of at this level of generality is summarised in [34]. The proofs of our
results, both with and without immigration, appeal to similar technical
approaches and accordingly, we include the results together in this paper.
