Collision-free navigation is a critical issue in robotic systems as the environment is often dynamic and uncertain. This paper investigates a data-stream-driven motion control problem for mobile robots to avoid randomly moving obstacles when the probability distribution of the obstacle’s movement is partially observable through data and can be even time-varying. A data-stream-driven ambiguity set is firstly constructed from movement data by leveraging a Dirichlet process mixture model and is updated online using real-time data. Then we propose an Online-Learning-based Distributionally Robust Nonlinear Model Predictive Control (OL-DR-NMPC) approach for limiting the risk of collision through considering the worst-case distribution within the ambiguity set. To facilitate solving the OL-DR-NMPC problem, we reformulate it as a finite-dimensional nonlinear optimization problem. To cope with the bilinear matrix inequality constraints in the nonlinear problem, we develop a parabolic relaxation and a sequential algorithm, by which the problem is further transformed into polynomial-time solvable surrogates. The simulations using a quadrotor model are employed to demonstrate the effectiveness and advantages of the proposed method.