This paper formulates the Sport Event Filming with Connectivity Constraints (SEF-C<inline-formula><tex-math notation="LaTeX">^2</tex-math> <mml:math><mml:msup><mml:mrow/><mml:mn>2</mml:mn></mml:msup></mml:math><inline-graphic xlink:href="zema-ieq1-2917176.gif"/> </inline-formula>) problem. The SEF-C<inline-formula><tex-math notation="LaTeX">^2</tex-math> <mml:math><mml:msup><mml:mrow/><mml:mn>2</mml:mn></mml:msup></mml:math><inline-graphic xlink:href="zema-ieq2-2917176.gif"/> </inline-formula> problem is an event coverage problem, which exploits a team of Unmanned Aerial Vehicles (UAVs) over a limited field in order to track the movements of an object (e.g., of the ball) and to deliver a video stream of the events (e.g., ball passes, goals) to the spectators meeting certain timeliness and video quality criteria. Assuming a priori knowledge of the whole sequence of actions, first a novel mathematical model is introduced that determines a sequence of movements for the UAVs, such that the timeliness of the filming is maximized and the total traveled distance is minimized. Then, dynamic, artificial potential function based, distributed UAV movement schemes that have no a priori knowledge of the sequence of game actions are proposed to optimize networking performance. Extensive simulations are used to analyze the performance in terms of video transmission quality and show that the proposed schemes outperform existing schemes.