Dynamic properties of grain boundaries play vital roles in the mechanical and plastic behaviors of polycrystalline materials. The properties of grain boundaries strongly depend on their microscopic structures. We present continuum models for the dynamics of grain boundaries based on the continuum distribution of the line defects (dislocations or disconnections) and the constraints associated them. The long-range elastic interaction between the line defects is included in the continuum models to maintain stable microstructure on grain boundaries during the evolution. However, the calculation of the long-range force is quite time-consuming due to its form of integrals over all the grain boundaries. This limitation can be addressed by replacing with constraints that governs the stable dislocation structure during the evolution. In the other hand, in polycrystalline materials, the motion of grain boundaries is inevitably constrained by other grains and triple junctions, and this essentially influences the materials properties. We incorporate these important microscopic constraints to continuum dynamics models of high-angle grain boundaries whose dynamics are controlled by motion of disconnections.