Wireless Sensor Networks (WSNs) technology is one of the building blocks ofthe Internet of Things (IoT). Due to their features of easy deployment and flexibility,they are used in many application domains. Low-Power and Lossy Networks(LLNs) are a special type of WSNs in which nodes are largely resources constrained.For LLNs, convergecast is one of the basic traffic modes, where all traffic in the networkis destined to a predefined destination called the sink. While considering theIoT application domains, convergecast is not the only traffic mode in the network.The sink needs to send commands to certain sensors to perform actions. In this application,anycast is another basic traffic mode. In anycast, the traffic from the sinkis destined to any member of a group of potential receivers in the network.Traditionally LLNs are formed by static sensor nodes and rarely change positions.Due to the strict resource constraints in computation, energy and memory ofLLNs, most routing protocols only support static network. However, mobility hasbecome an important requirement for many emerging applications. In these applications,certain nodes are free to move and organize themselves into a connectednetwork. The topology would continuously change due to the movement of nodesand radio links instability. This is a hard task for most routing protocols of LLNs toadapt rapidly to the movement and to reconstruct topology in a timely manner.The goal of this thesis is to propose an efficient mobility support for routingprotocols in LLNs. We focus on convergecast and anycast, which are the most usedtraffic modes in LLNs, in mobile network scenarios.We propose an enhancement mechanism, named RL (RSSI and Level), to supportrouting protocols in convergecast LLNs in mobility. This mechanism helps routingprotocol make faster decisions for detecting mobility and updating next-hop neighborsbut suffers from high overhead. We propose a dynamic control message managementto enhance the overhead performance of RL and implement it on top ofRouting Protocol for Low-power and Lossy network (RPL) and we named it RRD(RSSI, Rank and Dynamic). After taking into account hysteresis of the coveragezone of the transmission range of nodes, we optimized RRD. This enhanced versionis called RRD+. Based on RRD+, we proposed MRRD+ (Multiple, RSSI, Rankand Dynamic) to support multiple sinks in convergecast LLNs in mobility. ADUP(Adaptive Downward/Upward Protocol) is a routing solution that supports bothconvergecast and anycast in LLNs concurrently.We evaluated the performance of our contributions in both simulation usingCooja simulator and experiment (only for ADUP) on TelosB motes. The resultsobtained in both simulation and experiment confirm the efficiency of our routingprotocols.