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 performance-complexity 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.