We investigate the effect of three important processes by which active galactic nuclei (AGN)-blown bubbles transport material: drift, wake transport and entrainment. The first of these, drift, occurs because a buoyant bubble pushes aside the adjacent material, giving rise to a net upward displacement of the fluid behind the bubble. For a spherical bubble, the mass of upwardly displaced material is roughly equal to half the mass displaced by the bubble and should be ∼ 107–9 M⊙ depending on the local intracluster medium (ICM) and bubble parameters. We show that in classical cool-core clusters, the upward displacement by drift may be a key process in explaining the presence of filaments behind bubbles. A bubble also carries a parcel of material in a region at its rear, known as the wake. The mass of the wake is comparable to the drift mass and increases the average density of the bubble, trapping it closer to the cluster centre and reducing the amount of heating it can do during its ascent. Moreover, material dropping out of the wake will also contribute to the trailing filaments. Mass transport by the bubble wake can effectively prevent the buildup of cool material in the central galaxy, even if AGN heating does not balance ICM cooling. Finally, we consider entrainment, the process by which ambient material is incorporated into the bubble. Studies of observed bubbles show that they subtend an opening angle much larger than predicted by simple adiabatic expansion. We show that bubbles that entrain ambient material as they rise will expand faster than the adiabatic prediction; however, the entrainment rate required to explain the observed opening angle is large enough that the density contrast between the bubble and its surroundings would disappear rapidly. We therefore conclude that entrainment is unlikely to be a dominant mass transport process. Additionally, this also suggests that the bubble surface is much more stable against instabilities that promote entrainment than expect for pure hydrodynamic bubbles.