Interacting quantum many-body systems are expected to thermalize, in the sense that the evolution of local expectation values approaches a stationary value resembling a thermal ensemble. This intuition is notably contradicted in systems exhibiting many-body localisation (MBL). In stark contrast to the non-interacting case of Anderson localisation, the entanglement of states grows without limit over time, albeit slowly. In this work, we establish a novel link between quantum information theory and notions of condensed matterhysics, capturing thishenomenon in the Heisenbergicture. We show that the mere existence of local constants of motion, often taken as the definingroperty of MBL, together with a generic spectrum of the Hamiltonian, is already sufficient to rigorouslyrove informationropagation: these systems can be used to send a classical bit over arbitrary distances, in that the impact of a localerturbation can be detected arbitrarily far away. This counterintuitive result is compatible with and further corroborates the intuition of a slow entanglement growth following global quenches in MBL systems. Weerform a detailederturbation analysis of quasi-local constants of motion and also show that they indeed can be used to construct efficient spectral tensor networks, as recently suggested. Our resultsrovide a detailed and at the same time model-independenticture of informationropagation in MBL systems.