We will look at the problems of proper holomorphic mappings among certain domains on Grassmannians. These domains, known as flag domains, are the open orbits of the standard SU(p,q) actions on Grassmannians. Examples of flag domains include bounded symmetric domains (in particular, unit balls). However, in this talk we will focus on flag domains that contain compact complex submanifolds of positive dimension. We will see how the structures of these compact submanifolds can lead to rigidity of proper holomorphic maps among the domains. Our results cover certain cases of the theorem of Baouendi-Huang on generalized balls.