The non-linear evolution of one-dimensional perturbations in a three-dimensional expanding Universe is considered. A general Lagrangian scheme is derived, and compared to two previously introduced approximate models. These models are simulated with heap-based event-driven numerical procedure that allows for the study of large systems, averaged over many realisations of random initial conditions. One of the models is shown to be qualitatively, and, in some respects, concerning mass aggregation, quantitatively similar to the adhesion model.