Jet feedback from active galactic nuclei (AGN) harboured by brightest cluster galaxies is expected to play a fundamental role in regulating cooling in the intracluster medium (ICM). While observations and theory suggest energy within jet lobes balances ICM radiative losses, the modus operandi of energy communication with the ICM remains unclear. We present simulations of very high resolution AGN-driven jets launching in a live, cosmological galaxy cluster, within the moving mesh code arepo. As the jet propagates through the ICM the majority of its energy, which is initially in the kinetic form, thermalizes quickly through internal shocks and inflates lobes of very hot gas. The jets effectively heat the cluster core, with PdV work and weather-aided mixing being the main channels of energy transfer from the lobes to the ICM, while strong shocks and turbulence are subdominant. We additionally present detailed mock X-ray maps at different stages of evolution, revealing clear cavities surrounded by X-ray bright rims, with lobes being detectable for up to ∼108 yr even when magnetic draping is ineffective. We find bulk motions in the cluster can significantly affect lobe propagation, offsetting them from the jet direction and imparting bulk velocities that can dominate over the buoyantly rising motion.