We consider the problem of placing the minimum number of monitors in a dynamic network to identify additive link metrics from path metrics measured along cycle-free paths between monitors. Our goal is robust monitor placement, i.e., the same set of monitors can maintain network identifiability under topology changes. Our main contribution is a set of monitor placement algorithms with different performancecomplexity tradeoffs that can simultaneously identify multiple topologies occurring during the network lifetime. In particular, we show that the optimal monitor placement is the solution to a generalized hitting set problem, for which we provide a polynomial-time algorithm to construct the input and a greedy algorithm to select the monitors with logarithmic approximation. Although the optimal placement is NP-hard in general, we identify non-trivial special cases that can be solved efficiently. Our secondary contribution is a dynamic triconnected decomposition algorithm to compute the input needed by the monitor placement algorithms, which is the first such algorithm that can handle edge deletions. Our evaluations on mobility-induced dynamic topologies verify the efficiency and the robustness of the proposed algorithms.