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Macroscopically non-local effects are common in electron transport in semiconductor devices, occurring whenever the mean free path and/or the deBroglie wavelength are not small compared to geometry/flow length scales. When such effects are important, standard diffusion-drift (DD) theory becomes inaccurate and in need of revision, with the best known example being density-gradient (DG) theory wherein a gradient term is added to the electron gas equation of state to approximate the effect of quantum non-locality. Here we consider a similarly motivated gradient correction to the electron-lattice interaction that accounts for non-locality in the transport physics. Versions of DD and DG theory with this correction are discussed, and then are applied to the analysis of long-channel field-effect transistors where they are found to provide a physics-based approach for modeling and understanding interface scattering.