Named Data Networking (NDN) is a network architecture for the Future Internet, and can cooperate with the Internet of Things (IoT) to tackle problems existing in the current IP/TCP network architecture. Its consumer-driven paradigm and generic cache could optimize content distribution performance. However, there are still some problems to be dealt with, including mobility support of the producer. This paper proposes a producer mobility support scheme in NDN specifically for real-time multimedia delivery scenarios to reduce handover delay and to improve consumer experience with bounded overhead. Complementary components are designed based on forwarding information bases and packets in the original NDN. In particular, we categorize scenarios into two types according to cache state, and handle them with different strategies. Results of numerical analysis show that our scheme outperforms the original NDN and some typical solutions on delay with bounded overhead.