Many 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. Natural applications include epidemiology, individual-based modelling in ecology, spatio-temporal dynamics observed in bio-imaging, and computer vision. The aim of this article is to estimate in this context the birth and death intensity functions, that depend in full generality on the current spatial configuration of all alive individuals. While the temporal evolution of the population size is a simple birth-death process, observing the lifetime and trajectories of all individuals calls for a new paradigm. To formalise this framework, we introduce spatial birth-death-move processes, where the birth and death dynamics depends on the current spatial configuration of the population and where individuals can move during their lifetime according to a continuous Markov process with possible interactions.We consider non-parametric kernel estimators 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 the estimators in presence of continuous-time and 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, providing new insights on this complex mechanism.