We study a cellular networking scenario, called DroneCells, where miniaturized base stations (BSs) are mounted on flying drones to serve mobile users. We propose that the drones never stop, and move continuously within the cell in a way that reduces the distance between the BS and the serving users, thus potentially improving the spectral efficiency of the network. By considering the practical mobility constraints of commercial drones, we design drone mobility algorithms to improve the spectral efficiency of DroneCells. As the optimal problem is NP-hard, we propose a range of practically realizable heuristics with varying complexity and performance. Simulations show that, using the existing consumer drones, the proposed algorithms can readily improve spectral efficiency by 34\% and the 5-percentile packet throughput by 50\% compared to the scenario, where drones hover over fixed locations. More significant gains can be expected with more agile drones in the future. A surprising outcome is that the drones need to fly only at minimal speeds to achieve these gains, avoiding any negative effect on drone battery lifetime. We further demonstrate that the optimal solution provides only modest improvements over the best heuristic algorithm, which employs Game Theory to make mobility decisions for drone BSs.