Spatial birth-death processes are generalisations of simple birth-death processes, where the birth and death dynamics depend on the spatial locations of individuals. In this article, we further let individuals move during their life time according to a continuous Markov process. This generalisation, that we call a spatial birth-death-move process, finds natural applications in computer vision, bio-imaging and individual-based modelling in ecology. In a first part, we verify that birth-death-move processes are well-defined homogeneous Markov processes, we study their convergence to an invariant measure and we establish their underlying martingale properties. In a second part, we address the non-parametric estimation of their birth, death and total intensity functions, in presence of continuous-time or discrete-time observations. We introduce a kernel estimator that we prove to be consistent under fairly simple conditions, in both settings. We also discuss how we can take advantage of structural assumptions made on the intensity functions, and we explain how bandwidth selection by likelihood cross-validation can be conducted. A simulation study completes the theoretical results. We finally apply our model to the analysis of the spatio-temporal dynamics of proteins involved in exocytosis mechanisms in cells.