Internet of Mobile Things (IoMT) is a new paradigm of the Internet of Things (IoT) where devices such as sensors, robots, unmanned aerial vehicles (UAV) and cars, are inherently mobile. While mobility enables innovative applications and allows new services, it remains a challenging issue as it causes disconnection of nodes and intermittent connectivity, which negatively impact the network performance; namely data loss, large handover delay and application functionality failures. In this paper, we propose a new energy efficient and mobility aware routing protocol named EC-MRPL based on the well-known Routing Protocol for Low power and Lossy Networks (RPL standard). Unlike RPL which is designed for low resources networks with basically static devices, the proposed protocol enables to better conserve the energy and sustain the connectivity of mobile nodes. EC-MRPL integrates an enhanced mobility detection method and a novel point of attachment prediction and replacement strategy aware of the resources constraints. As such, EC-MRPL overcomes and mitigates problems caused by mobility. Obtained simulation results using Cooja/Contiki show that EC-MRPL outperforms both the RPL and the MRPL protocols in terms of handover delay, data loss rate, signaling cost and energy consumption.