Generators of space-time dynamics in bioimaging have become essential to build ground truth datasets for image processing algorithm evaluation such as biomolecule detectors and trackers, as well as to generate training datasets for deep learning algorithms. In this contribution, we leverage a stochastic model, called birth-death-move (BDM) point process, in order to generate joint dynamics of biomolecules in cells. This approach is very flexible and allows us to model a system of particles in motion, possibly in interaction, that can each possibly switch from a motion regime (e.g. Brownian) to another (e.g. a directed motion), along with the appearance over time of new trajectories and their death after some lifetime, all of these features possibly depending on the current spatial configuration of all existing particles. We explain how to specify all characteristics of a BDM model, with many practical examples that are relevant for bioimaging applications. Based on real fluorescence microscopy datasets, we finally calibrate our model to mimic the joint dynamics of Langerin and Rab11 proteins near the plasma membrane. We show that the resulting synthetic sequences exhibit comparable features as those observed in real microscopy image sequences.