Chalcogenide-based phase-change materials play a prominent role in information technology. In spite of decades of research, the details of electrical transport in these materials are still debated. In this article, we present a unified model based on multiple-trapping transport together with 3D Poole-Frenkel emission from a two-center Coulomb potential. With this model, we are able to explain electrical transport both in as-deposited phase-change material thin films, similar to experimental conditions in early work dating back to the 1970s, and in melt-quenched phase-change materials in nanometer-scale phase-change memory devices typically used in recent studies. Experimental measurements on two widely different device platforms show remarkable agreement with the proposed mechanism over a wide range of temperatures and electric fields. In addition, the proposed model is able to seamlessly capture the temporal evolution of the transport properties of the meltquenched phase upon structural relaxation.