Grid Services became a widely used technology for building service oriented applications. The Web Service Resource Framework (WSRF) is the current standard used for building such services as it describes the way to design and communicate with stateful services. However, deploying Web services on a grid environment raises many challenges such as adapting the Web service to the dynamic change of grid resources performance and availability. Therefore, mechanisms, such as service mobility, may be very helpful for supporting autonomic properties of grid services. In this paper, we propose a solution for WSRF grid services strong mobility allowing services to migrate during their execution while keeping their state consistent. In addition, suspended services will resume their execution starting from the interruption point. To show the feasibility of our approach, we present a case study illustrating the application of our transformation on a WSRF Auction service. We provide then some preliminary evaluations performed on the Globus Toolkit (version4.0).