jats:pWe investigate the transport of relatively heavy, small particles through a downward displacement-ventilated space. A flux of particles is supplied to the space from a localised source at a high level and forms a turbulent particle-laden plume which descends through the space. A constant flow of ambient fluid which does not contain particles is supplied to the space at a high level, while an equal amount of fluid is vented from the space at a low level. As a result of the entrainment of ambient fluid into the particle plume, a return flow is produced in the ambient fluid surrounding the plume in the lower part of the space. At steady state, particles are suspended by this return flow. An interface is formed which separates the ambient fluid in the lower part of the space, which contains particles, from the particle-free ambient fluid in the upper part of the space. New laboratory experiments show that the concentration of particles in the ambient fluid below the interface is larger than the average concentration of particles in the plume fluid at the level of the interface. Hence, as the plume fluid crosses the interface and descends through the particle-laden fluid underneath, it becomes relatively buoyant and forms a momentum-driven fountain. If the fountain fluid impinges on the floor, it then spreads radially over the surface until lifting off. We develop a quantitative model which can predict the height of the interface, the concentration of particles in the lower layer, and the partitioning of the particle flux between the fraction which sediments over the floor and that which is ventilated out of the space. We generalise the model to show that when particles and negatively buoyant fluid are supplied at the top of the space, a three-layer stratification develops in the space at steady state: the upper layer contains relatively low-density ambient fluid in which no particles are suspended; the central layer contains a mixture of ambient and plume fluid in which no particles are suspended; and the lower layer contains a suspension of particles in the same mixture of ambient and plume fluid. We quantify the heights of the two interfaces which separate the three layers in the space and the concentration of particles in suspension in the ambient fluid in the lower layer. We then discuss the relevance of the results for the control of airborne infections in buildings. Our experiments show that the three-layer stratification is subject to intermittent large-scale instabilities when the concentration of particles in the plume at the source is sufficiently small, or the rate of ventilation of the space is sufficiently large: we describe the transient concentration of particles in the space during one of these instabilities.</jats:p>