Network slicing was adopted as a solution for the future mobile network to support various applications with diverse requirements. While active research has focused on this functionality, most of the work targets RAN or packet core slicing and leaves the transport network nearly untouched. With packet core functions moving to data centers and parts of RAN functions moving to edge clouds, the transport network will gain significantly more importance. To ensure tight service level agreements (SLAs) and facilitate slice management, it is imperative for the transport network to support live slice mobility with no disruption of current services. In this paper, we present FestNet, a Flexible and Efficient Sliced Transport Network, enabled by our virtualized Programmable Data Plane (vPDP) and two-layer design. Not only protocols but also stateless and stateful network functions (NFs) can be run as slices in FestNet. We implemented a FestNet prototype. Our evaluation shows that FestNet supports live slice migration with no packet loss and no state loss for stateful slices and that FestNet provides many times faster slice operations than implementations in related work.