We consider the problem of an agent/robot with non-holonomic kinematics avoiding many dynamic obstacles. State and velocity noise of both the robot and obstacles as well as the robot's control noise are modelled as non-parametric distributions as often the Gaussian assumptions of noise models are violated in real-world scenarios. Under these assumptions, we formulate a robust MPC that samples robotic controls effectively in a manner that aligns the robot to the goal state while avoiding obstacles under the duress of such non-parametric noise. In particular, the MPC incorporates a distribution matching cost that effectively aligns the distribution of the current collision cone to a certain desired distribution whose samples are collision-free. This cost is posed as a distance function in the Hilbert Space, whose minimization typically results in the collision cone samples becoming collision-free. We compare and show tangible performance gain with methods that model the collision cone distribution by linearizing the Gaussian approximations of the original non-parametric state and obstacle distributions. We also show superior performance with methods that pose a chance constraint formulation of the Gaussian approximations of non-parametric noise without subjecting such approximations to further linearizations. The performance gain is shown both in terms of trajectory length and control costs that vindicates the efficacy of the proposed method. To the best of our knowledge, this is the first presentation of non-holonomic collision avoidance of moving obstacles in the presence of non-parametric state, velocity and actuator noise models.