In a wireless sensor network (WSN), finding the optimal route from each node to the sink is not a straightforward task because of the distributed and dynamic characteristics of the network. For instance, the network suffers frequent changes because the channel quality varies over time and the nodes can leave or join the network at any moment. In order to deal with this variability, we propose the Dynamic Gallager-Humblet-Spira (DGHS) algorithm that builds and maintains a minimum spanning tree for distributed and dynamic networks, and we have found that DGHS reduces the number of control messages and the energy consumption, at the cost of a slight increase in the memory size and convergence time. This paper presents a detailed description of the different stages of the DGHS algorithm (neighbor discovery, tree construction and data collection), an in-depth analysis of extensive results that validates our proposal, and an exhaustive description of the GHS limitations.