The ability of cells to undergo collective movement plays a fundamental role in tissue repair, development and cancer. Interactions occurring at the level of individual cells may give rise to spatial structure, such as clustering, in a moving population. In vitro cell culture studies have shown that the presence of such spatial structure can play an important role in determining the dynamics of migrating cells at a population level. However, mathematical models that consider population-level behaviour often take a mean-field approach, which assumes that individuals interact with one another in proportion to their average density and neglects the effects of spatial structure. In this work, we develop a lattice-free individual-based model (IBM) for collective movement in one-dimensional space. The IBM uses random walk theory to model the stochastic interactions occurring at the scale of individual migrating cells. In particular, our model allows an individual's direction of movement to be affected by interactions with other cells in its neighbourhood, providing insights into how directional bias generates spatial structure. As an alternative to the mean-field approach, we employ spatial moment theory to develop a population-level model which accounts for spatial structure and predicts how these individual-level interactions propagate to the scale of the whole population. The IBM is used to derive an equation for dynamics of the second spatial moment (the average density of pairs of cells) which incorporates the neighbour-dependent directional bias and we solve this numerically for a spatially homogeneous case. Extending our model to consider cell behaviour in two-dimensional space makes it more amenable for use alongside experimental data. Using imaging data from in vitro experiments, we estimate parameters for the two-dimensional model and show that it can generate similar spatial structure to that observed in a 3T3 fibroblast cell population. Finally, we incorporate cell birth and death into our two-dimensional model to consider how these processes give rise to spatial structure and how, in turn, this spatial structure affects the collective dynamics.