We analyze the propagation of excitons in a d-dimensional lattice with power-law hopping ∝1/r^α in the presence of dephasing, described by a generalized Haken-Strobl-Reineker model. We show that in the strong dephasing (quantum Zeno) regime the dynamics is described by a classical master equation for an exclusion process with long jumps. In this limit, we analytically compute the spatial distribution, whose shape changes at a critical value of the decay exponent α_cr=(d+2)/2. The exciton always diffuses anomalously, a superdiffusive motion is associated to a Lévy stable distribution with long-range algebraic tails for α≤α_cr, while for α>α_cr the distribution corresponds to a surprising mixed Gaussian profile with long-range algebraic tails, leading to the coexistence of short-range diffusion and long-range Lévy-flights. In the many-exciton case, we demonstrate that, starting from a domain-wall exciton profile, algebraic tails appear in the distributions for any α, which affects thermalization such that the longer is the hopping range, the faster equilibrium is reached. Our results are directly relevant to experiments with cold trapped ions, Rydberg atoms and supramolecular dye aggregates. They provide a way to realize an exclusion process with long jumps experimentally.