Various spatio-temporal data record the time of birth and death of individuals, along with their spatial trajectories during their lifetime, whether through continuous-time observations or discrete-time observations. The data at hand can be viewed as a random set of points, the cardinality and the position of which evolve stochastically through time. Natural applications include epidemiology, individual-based modelling in ecology, spatio-temporal dynamics observed in bio-imaging, and computer vision. To model this kind of data, we introduce spatial birth-death-move processes, where the birth and death dynamics depends on the current spatial state of all alive individuals and where individuals can move during their lifetime according to a continuous Markov process. We present some of the basic probabilistic properties of these processes and we consider the non-parametric estimation of their birth and death intensity functions. The setting is original because each observation in time belongs to a non-vectorial, infinite dimensional space and the dependence between observations is barely tractable. We prove the consistency of kernel estimators in presence of continuous-time or discrete-time observations, under fairly simple conditions. We moreover discuss how we can take advantage in practice of structural assumptions made on the intensity functions and we explain how data-driven bandwidth selection can be conducted, despite the unknown (and sometimes undefined) second order moments of the estimators. We finally apply our statistical method to the analysis of the spatio-temporal dynamics of proteins involved in exocytosis in cells. This is a joint work with Ronan Le Guével (Rennes 2). The associated article is available here and its preprint version here