In Future Space-Terrestrial Integrated Networks (FSTINs), mobility is the norm rather than the exception, the current TCP/IP architecture is not competent. As a promising future network architecture, Named Data Networking (NDN) can support content consumer mobility naturally, but the content producer mobility support remains a challenging problem. Most previous research simply considered this problem in terrestrial scenarios, which involve stable infrastructures to achieve node mobility management. In this paper, we consider the problem in an FSTIN scenario without special handover management infrastructures. Specifically, we propose a tracing-based producer mobility management scheme and an addressing-assisted forwarding method via NDN architecture. We formally describe Multi-Layered Satellite Networks via a Time Varying Graph model and define the foremost path calculating problem to calculate the route of space segment, as well as an algorithm that can function in both dense (connected) and sparse (delay/disruption tolerant) scenarios. Finally, we discuss the acceleration method that can improve the Space-Terrestrial Integrated forwarding efficiency. Performance evaluation demonstrates that the proposed scheme can support fast handover and efficient forwarding in the FSTIN scenario.