Considered is a mobile ad hoc network consisting of three types of nodes (source, destination and relay nodes) and using the two-hop relay routing protocol. Packets at relay nodes are assumed to have a limited lifetime in the network. All nodes are moving inside a bounded region according to some random mobility model. Both closed-form expressions, and asymptotic results when the number of nodes is large, are provided for a number of QoS metrics related to the packet delivery delay and the energy needed to transmit a packet from the source to its destination. Markov models and mean-field approximations are used. We also introduce and evaluate a variant of the two-hop relay protocol that limits the number of copies of a packet in the network. Our model is validated through simulations for two mobility models (random waypoint and random direction mobility models), numerical results for the two-hop relay protocols are reported, and the performance of the two-hop routing and of the epidemic routing protocols are compared.