Jets from supermassive black holes in the centres of galaxy clusters are a potential candidate for moderating gas cooling and subsequent star formation through depositing energy in the intra-cluster gas. In this work, we simulate the jet-intra-cluster medium interaction using the moving-mesh magnetohydrodynamics code Arepo. Our model injects supersonic, low density, collimated and magnetised outflows in cluster centres, which are then stopped by the surrounding gas, thermalise and inflate low-density cavities filled with cosmic-rays. We perform high-resolution, non-radiative simulations of the lobe creation, expansion and disruption, and find that its dynamical evolution is in qualitative agreement with simulations of idealised low-density cavities that are dominated by a large-scale Rayleigh-Taylor instability. The buoyant rising of the lobe does not create energetically significant small-scale chaotic motion in a volume-filling fashion, but rather a systematic upward motion in the wake of the lobe and a corresponding back-flow perpendicular to it. We find that, overall, 50 per cent of the injected energy ends up in material which is not part of the lobe, and about 25 per cent remains in the inner 100 kpc. We conclude that jet-inflated, buoyantly rising cavities drive systematic gas motions which play an important role in heating the central regions, while mixing of lobe material is subdominant. Encouragingly, the main mechanisms responsible for this energy deposition can be modelled already at resolutions within reach in future, high-resolution cosmological simulations of galaxy clusters.