The joint localization and synchronization (JLAS) of moving agents with clock offsets is critical to facilitating location services for Internet of Things (IoT). Existing methods using one-way time-of-arrival measurements require anchors to be synchronized and the agent's motion to be modeled with a constant velocity. However, the requirement of synchronization between anchors limits the flexibility and scalability of IoT networks. Moreover, existing methods are inapplicable for agents that perform arbitrary motions. In this study, we developed a set of methods to solve the JLAS problem in asynchronous networks. First, we modeled the motion of the moving agent and classified it into two categories: 1) acceleration and 2) time-varying motion. We showed that the existing modeling motion is a special case of acceleration motion. Second, for the case of acceleration motion with priori information, we proposed the optimal JLAS method, namely, JLAS-KAM, to compensate for the movement-caused estimation error. Next, we developed a maximum likelihood estimator, namely, JLAS-UAM, to jointly estimate the position, velocity, and acceleration of the agent in the absence of priori information. Third, for the case of time-varying motion, we developed the optimal JLAS method, namely, JLAS-TVM, to jointly estimate the agent position and velocity at each time instant. Moreover, iterative algorithms were proposed to solve optimization problems. We derived the Cramer-Rao lower bound for three proposed methods and analyzed their performance. Simulation results verified the theoretical analysis of the estimation performance and revealed the characteristics and advantages of the proposed methods.