Despite the ubiquity of applications of heat transport across nanoscale interfaces, including integrated circuits, thermoelectrics, and nanotheranostics, an accurate description of phonon transport in these systems remains elusive. Here we present a theoretical and computational framework to describe phonon transport with position, momentum, and scattering event resolution. We apply this framework to a single-material nanoparticle for which this multidimensional resolution offers insight into the physical origin of phonon thermalization and the length-scale dependent anisotropy of driven phonon distributions. We extend the formalism to handle interfaces and investigate the specific case of semicoherent materials interfaces by computing the coupling between phonons and interfacial strain resulting from a periodic array of misfit dislocations. We calculate the thermal interface conductance within the technologically relevant Si-Ge heterostructures and obtain G=173.2MW/m2K, in good agreement with previous experimental and theoretical work. Finally we comment on future applications of our framework including coherent and driven phonon effects in nanoscale materials, which are increasingly accessible via ultrafast, terahertz, and near-field spectroscopies.