Content-centric networking (CCN) is designed for efficient dissemination of information. Several architectures are proposed for CCN recently, but mobility issues are not considered sufficiently. We classify traffic types of CCN into real-time and non real-time. We examine mobility problems for each type, and suggest the possible hand-off schemes over CCN. Then, we analyze the delay performance in terms of simulation study. We believe that the proposed schemes can be merged as a part of the CCN easily, since they comply with the inherent nature and rules of the CCN.