The emerging drone swarms are capable of carrying out sophisticated tasks in support of demanding Internet-of-Things (IoT) applications by synergistically working together. However, the target area may be out of the coverage of the ground station and it may be impractical to deploy a large number of drones in the target area due to cost, electromagnetic interference and flight-safety regulations. By exploiting the innate \emph{agility} and \emph{mobility} of unmanned aerial vehicles (UAVs), we conceive a mobile relaying-assisted drone swarm network architecture, which is capable of extending the coverage of the ground station and enhancing the effective end-to-end throughput. Explicitly, a swarm of drones forms a data-collecting drone swarm (DCDS) designed for sensing and collecting data with the aid of their mounted cameras and/or sensors, and a powerful relay-UAV (RUAV) acts as a mobile relay for conveying data between the DCDS and a ground station (GS). Given a time period, in order to maximize the data delivered whilst minimizing the delay imposed, we harness an $ε$-multiple objective genetic algorithm ($ε$-MOGA) assisted Pareto-optimization scheme. Our simulation results demonstrate that the proposed mobile relaying is capable of delivering more data. As specific examples investigated in our simulations, our mobile relaying-assisted drone swarm network is capable of delivering $45.38\%$ more data than the benchmark solutions, when a stationary relay is available, and it is capable of delivering $26.86\%$ more data than the benchmark solutions when no stationary relay is available.