We consider a dynamic game model of ride-sourcing, where a large number of private car owners provide rides to randomly appearing customers. Free drivers can travel around the city to improve their chance of being hired. They have access to the origin-destination statistics, to the current customer requests and to information about traffic congestion and the time- varying connectivity of the road network. We show how each driver can compute a best- response strategy to the anticipated behavior of the other drivers, which leads to an approximate Nash equilibrium in the limit of an infinite number of players. The outputs of our discrete-time model are the cars' individual paths and the distribution of the free and busy drivers at each period. Finally, a numerical example illustrates three scenarios where a road closure in a city makes the cars desert an area. It shows how a financial incentive, namely increasing the ride fare in the deserted area, helps reestablish the ride service in that region.