The process of a particle, solute in matrix, solute/impurities at boundaries, etc. interacting with the boundary to impede or slow its motion is one that molecular dynamics (and/or mesoscale simulations) can shed some light on.  On the molecular dynamics front, there has been a lot of work in synthetic driving forces, but these may not be compatible with binary interatomic potentials yet.  Some people (e.g., Mishin) have looked at grain boundary motion in the presence of solutes via Monte Carlo techniques for problems of thermal stability in nanocrystalline materials, for instance.  The ability to couple an evolving growth process for particles via phase field has recently (within the last 5 years or so) been coupled with Potts Monte Carlo to generate more efficient ways to couple grain growth in polycrystalline samples with diffusional/compositional physics.  In terms of mathematical models, as you mentioned, there are some basic ones out there and it would be good to tie in insight from the aforementioned modeling and simulations to come up with more physics-based mathematical models.

Along a similar line of thought, the interaction energy with grain boundaries can be easily computed at the molecular dynamics or quantum mechanics scales.  The formation energy is a well known concept, but the binding energy can describe the additional energy involved with separating/combining two defects, i.e., two solutes, a solute and grain boundary, an interstitial He and a growing He bubble, etc.  I attached a couple of slides from a conference presentation that I am giving in a few days, if you are interested.  A simple mathematical model formulation was used to describe the binding energies between defects/solutes/impurities, but it is obvious that local variations in the sites within the grain boundary can have a big impact, so a physically-based model would have to account for these.