Grain boundary processes such as shear coupling and sliding are a consequence of plastic distortion accompanying grain boundary motion. Atomic scale studies using molecular dynamics (MD) and phase field crystal simulations demonstrate non-unique and stress-dependent shear coupling, a key feature that recently motivated the introduction of disconnections as the primary carriers of grain boundary plasticity. In this work, we develop a diffuse-interface finite deformation theory for grain boundary plasticity based on the notion of geometrically necessary disconnections (GND), with a discrete collection of intrinsic coupling factors, defined by the bicrystallography, as inputs to the model. It is emphasized that disconnection density is not a primary variable of the theory, instead it is the plastic slip rate associated with shear coupling that is a fundamental variable, and the resulting incompatibility is measured by the GND density. As an interesting consequence of this framework, we discover that in the presence of grain rotation, plastic shear that is commonly associated with the glide of disconnections, is insufficient to describe grain boundary plasticity, and it must be augmented with an additional plastic rotation. The disconnections-based continuum model can describe phenomena such as state-dependent shear coupling, mode switching, grain boundary sliding and grain rotation. Using finite elements, we simulate grain boundaries driven by shear and synthetic driving forces in bicrystals under various boundary conditions. © 2022