The IoT devices with embedded mobile devices create the Internet of Mobile Things (IoMT) paradigm. Mobility is not supported by the routing protocol for low-power and lossy networks (RPL) created for static networks. IoMT has raised routing challenges such as link failure, instability, energy depletion, packet loss, and handover delay in the network. In this context, IoMT Fuzzy-based RPL (IoMT-FRPL) is proposed in this paper to enhance routing performance. Receiving Signal Strength Indicator (RSSI), Euclidean distance, Hop Count, and Expected Transmission Count (ETX) metrics are built into the fuzzy interface system for the mobile nodes in the network to conserve energy. The IoMT-FRPL consists of the following three key steps: The first steps are data transmission and motion investigation, the second is fuzzy-based prediction of a new static parent for the mobile node, and the third is verifying the unique attachment point. When conventional RPL, mRPL, and EMA-RPL were compared to IoMT-performance FRPL's in Cooja/Contiki 2.7, the simulation results revealed improvements in energy consumption, handover delay, packet delivery rate (PDR), and signaling cost.