The Internet of Things (IoT) can avail from device-to-device (D2D) communication techniques to increase object data exchange performance. IoT networks aim to offer a massive number of services at high quality levels, and many of the devices providing these services are mobile. Devices such as wearables, sensors, drones and smart vehicles need constant connectivity despite their moving patterns and therefore, an IoT architecture should consider both Quality of Service (QoS) and mobility. D2D allows devices to communicate directly to share content and functionality, such as access to the Internet. This paper proposes REMOS-IoT - A RElay and MObility Scheme for improved IoT communication performance in support of increased QoS for the data exchange services between mobile IoT devices. Simulation-based testing showed how performance of devices increased in several scenarios, demonstrating the efficiency of the proposed architecture and algorithms.