We develop two statistical models for space-time abundance data based on a stochastic underlying continuous individual movement. In contrast to current models for abundance in statistical ecology, our models exploit the explicit connection between movement and counts, including the induced space-time auto-correlation. Our first model, called Snapshot, describes the counts of free moving individuals with a false-negative detection error. Our second model, called Capture, describes the capture and retention of moving individuals, and it follows an axiomatic approach based on three simple principles from which it is deduced that the density of the capture time is the solution of a Volterra integral equation of the second kind. Mild conditions are imposed to the underlying stochastic movement model, which is free to choose. We develop simulation methods for both models. The joint distribution of the space-time counts provides an example of a new multivariate distribution, here named the evolving categories multinomial distribution, for which we establish key properties. Since the general likelihood is intractable, we propose a pseudo-likelihood fitting method assuming multivariate Gaussianity respecting mean and covariance structures, justified by the central limit theorem. We conduct simulation studies to validate the method, and we fit our models to experimental data of a spreading population. We estimate movement parameters and compare our models to a basic ecological diffusion model. Movement parameters can be estimated using abundance data, but one must be aware of the necessary conditions to avoid underestimation of spread parameters.