In this paper, we consider a status updating system where the transmitter sends status updates of the signal it monitors to the destination through a rate-limited link. We consider the scenario where the status of the monitored signal only changes at discrete time points. The objective is to let the destination be synchronized with the source in a timely manner once a status change happens. What complicates the problem is that the transmission takes multiple time slots due to the link-rate constraint. Thus, the transmitter has to decide to switch or to skip a new update when the status of the monitored signal changes and it has not completed the transmission of the previous one yet. We adopt a metric called 'Age of Synchronization' (AoS) to measure the 'dissatisfaction' of the destination when it is desynchronized with the source. Then, the objective of this paper is to minimize the time-average AoS by designing optimal transmission policies for the transmitter. We formulate the problem as a Markov decision process (MDP) and prove the multi-threshold structure of the optimal policy. Based on that, we propose a low computational-complexity algorithm for the MDP value iteration. We then evaluate the performance of the multi-threshold policy through simulations and compare it with two baseline policies and the AoI-optimal policy.