The intrusion of a fixed volume of fluid which is released from rest and then propagates horizontally at the neutral buoyancy level in a vertically stratified ambient fluid is investigated. The density change is linear, in a restricted layer or over the full depth of the container, and locks of both rectangular and cylindrical shapes are considered. A closed one-layer shallow-water inviscid formulation is used to obtain solutions of the initial-value problem. Similarity solutions for the large-time developed motion and an approximate box model are also presented. The results are corroborated by numerical solutions of the full two-dimensional Navier–Stokes equations and comparisons with previously published experiments. It is shown that the model is a versatile predictive tool which clarifies essential features of the flow field. Accurate insights are provided concerning the governing dimensionless parameters and the major features of the motion. In particular, the theory predicts and explains: ($a$) the fact that the initial propagation is with constant speed for intrusions released from a rectangular lock; ($b$) the effect of the shape of the lock on the motion; ($c$) the spread with time at some power in the developed stage; and ($d$) the sub-critical (compared to the mode 2 linear waves) speed in a full-depth stratified container configuration. The main deficiency of the shallow-water model is that internal gravity waves are not incorporated, but some insight into this effect is provided by the comparisons with the Navier–Stokes simulations and experiments.