In North America, any man's car can run on any man's railroad. The exception to this is the narrow gauge lines, which is a whole different subject. If a shipper in Denver wanted to ship a car to a consignee in Atlanta, the car might be routed over UP from Denver to Memphis and then go over the NS from Memphis to Atlanta via Sheffield and Birmingham. Since UP is the originating railroad in our example, they would most likely own the car. NS would bill them a certain amount to move the car over their railroad from Memphis to Atlanta. This example is highly simplified but it should help you understand how the interchange system works.
George