Avoiding collisions with obstacles is of fundamental importance for the safe navigation of unmanned aerial vehicles (UAVs) and mobile robots. In this paper, we approach the avoidance problem by composing a scalable navigation strategy from multiple stochastic optimal controllers. We consider a scenario with a fixed speed Dubins vehicle, which is tasked to reach a waypoint while avoiding collisions with multiple moving obstacles. Obstacle moving directions are unknown, therefore, we use a random walk stochastic process model to anticipate that uncertainty in the design of navigation feedback control. The proposed navigation is based on a composition of minimum time stochastic optimal controllers. Each optimal controller is the solution to a minimum time problem to reach either the waypoint or a safe configuration with respect to an obstacle. The composition is based on the controller value functions and is scalable, i.e., it can deal with any number of obstacles. Our results are illustrated with a numerical simulation.