Transport is an archetypical example of complex non-equilibrium phenomena impacting essentially all areas of physics. This theoretical work was motivated by the lack of a good theoretical description of our recent experiments on dipolar energy transport in a gas of atoms optically-coupled to short-lived states acting as a reservoir [related experiments: Günter et al, Science 2013]. We investigated the transport of excitations through a chain of atoms with nonlocal dissipation introduced through coupling to additional short-lived states. We found the system can be described by an effective spin-1/2 model where the ratio of the exchange interaction strength to the reservoir coupling strength determines the type of transport, including coherent exciton motion, incoherent hopping, and a regime in which an emergent length scale leads to a preferred hopping distance far beyond nearest neighbors. For multiple impurities, in addition to the immediate applications for introducing and controlling new types of transport, we discover this system may also serve as a concrete realisation of recent proposals for entanglement creation through dissipative coupling to engineered reservoirs.