I will present 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 movements. 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. Based on a joint work with Omer Bobrowski and Robert J. Adler.