We analyze the achievable data rate of cooperative relaying strategies in networks where nodes operate in half-duplex mode. Nodes have to deliver their data to a gateway, at a certain rate, and may have limited energy capabilities, as in the case of energy-harvesting communication networks. Both the requested data rate and the available energy capabilities may vary from node to node. Under such constraints, we take an information-theoretic approach and derive cut-set upper bounds to the achievable rate. Furthermore, we devise two kinds of communication strategies, each aiming at a different objective. The former ensures a fair rate allocation to the network nodes, but it neglects their energy constraints. The latter does consider energy constraints by meeting the requirements on the average power consumption at each node and by providing fairness in the data rate allocation. We show the performance of the aforementioned communication strategies, highlighting their effectiveness and providing useful insights on the system behavior.