Transfer of excitation energy is a key step in light harvesting and hence of technological relevance for solar energy conversion. In bare organic materials energy transfer proceeds via incoherent hops, which restrict propagation lengths to nanometers. In contrast, energy transport over several micrometers has been observed in the strong coupling regime where excitations hybridise with confined light modes to form polaritons. Because polaritons have group velocity, their propagation should be ballistic and long-ranged. However, experiments indicate that organic polaritons propagate in a diffusive manner and more slowly than their group velocity. Here, we resolve this controversy by means of molecular dynamics simulations of Rhodamine molecules in a Fabry-Pérot cavity. Our results suggest that polariton propagation is limited by the cavity lifetime and appears diffusive due to reversible population transfers between bright polaritonic states that propagate ballistically at their group velocity, and dark states that are stationary. Furthermore, because long-lived dark states transiently trap the excitation, propagation is observed on timescales beyond the intrinsic polariton lifetime. These atomistic insights not only help to better understand and interpret experimental observations, but also pave the way towards rational design of molecule-cavity systems for achieving long-range coherent energy transport.