Autonomous drone racing requires powerful perception, planning, and control and has become a benchmark and test field for autonomous, agile flight. Existing work usually assumes static race tracks with known maps, which enables offline planning of time-optimal trajectories, performing localization to the gates to reduce the drift in visual-inertial odometry (VIO) for state estimation or training learning-based methods for the particular race track and operating environment. In contrast, many real-world tasks like disaster response or delivery need to be performed in unknown and dynamic environments. To make drone racing more robust against unseen environments and moving gates, we propose a control algorithm that operates without a race track map or VIO, relying solely on monocular measurements of the line of sight to the gates. For this purpose, we adopt the law of proportional navigation (PN) to accurately fly through the gates despite gate motions or wind. We formulate the PN-informed vision-based control problem for drone racing as a constrained optimization problem and derive a closed-form optimal solution. Through simulations and real-world experiments, we demonstrate that our algorithm can navigate through moving gates at high speeds while being robust to different gate movements, model errors, wind, and delays.