We study the dynamics of the entanglement structure of a multipartite system experiencing a dissipative evolution. We characterize the processes leading to a particular form of output-system entanglement and provide a recipe for their identification via concatenations of particular linear maps with entanglement-breaking operations. We illustrate the applicability of our approach by considering local and global depolarizing noises acting on general multiqubit states. A difference in the typical entanglement behavior of systems subjected to these noises is observed: the originally genuine entanglement dissociates by splitting off particles one by one in the case of local noise, whereas intermediate stages of entanglement clustering are present in the case of global noise. We also analyze the definitive phase of evolution when the annihilation of the entanglement compound finally takes place.