Recent advancements in autonomous mobile robots (AMRs), such as aerial drones, ground vehicles, and quadrupedal robots, have significantly impacted the fields of infrastructure inspection, emergency response, and surveillance. Many of these settings contain multiple moving elements usually neglected in the planning process. While a large body of work covers topics addressing scenarios with stationary objects, promising work with dynamic points of interest has only recently gained traction due to computational complexity. The nature of the problem brings with it the challenges of motion prediction, real time adaptability, efficient decision-making, and uncertainty. Concerning aerial drones, while significantly constrained computationally, good understanding and the relative simplicity of their platform gives way to more complex prediction and planning algorithms needed to work with multiple moving objects. This paper presents a survey of the current state-of-the-art solutions to the path planning problem for multiple moving object inspection using aerial drones. The presented algorithms and approaches cover the challenges of motion and intention prediction, obstacle avoidance, planning in dynamic environments, as well as scenarios with multiple agents. Potential solutions and future trends were identified primarily in the form of heuristic and learning methods, state-of-the-art probabilistic prediction algorithms, and further specialization in regard to every scenario.