  We establish functional limit theorems for local, additive, interaction
functions of temporally evolving point processes. The dynamics are those of a
spatial Poisson process on the flat torus with points subject to a birth-death
mechanism, and which move according to Brownian motion while alive. The results
reveal the existence of a phase diagram describing at least three distinct
structures for the limiting processes, depending on the extent of the local
interactions and the speed of the Brownian motions. The proofs, which identify
three different limits, rely heavily on Malliavin-Stein bounds on a
representation of the dynamic point process via a distributionally equivalent
marked point process.
