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 melt-quenched phase upon structural relaxation.