Swarms of Unmanned Aerial Vehicles (UAVs) are a key technology to support communication in many harsh environments where fixed infrastructures (e.g., 5G) are disrupted or not available. However, the fast mobility and highly dynamic network topology pose unique challenges and require the development of novel multi-hop routing protocols. Previous work in this direction extends geographical protocols or adapts approaches designed for Mobile Ad-hoc NETworks (MANETs), rarely taking full advantage of UAV capabilities. In this paper, we introduce a novel data offloading approach, namely Stop & Offload, that exploits the device controllable mobility to facilitate network routing. The swarm of UAVs performs data offloading synchronously and recurrently. At fixed intervals of time, the swarm interrupts the sensing mission (Stop) and moves, as little as possible, to build a connected formation to the base station and offload the data (Offload). We provide both centralized solutions — assuming a long-range control channel — and a distributed solution — working in the absence of a control channel. By means of extensive simulations we show that our proposals outperform state-of-the-art solutions, decreasing the time taken to build a connected formation of about 45% and increasing the time spent on sensing of 10%. Additionally, we compared our protocol with various routing strategies and observe remarkable improvements, including a 50% reduction in average packet delay.