"Continuing flight numbers" are a US airline feature not found elsewhere in the world.

The greatest frauds are those where there is an actual change to a different type of aircraft at the intermediate point, which if the incoming aircraft is late arriving can get dispatched on time and all the "through" passengers get thrown out halfway along their supposed through trip. This used to be a regular feature of Pan Am and TWA in Europe in the 1980s, with one through flight number from a 747 across the Atlantic onto 727s operating to secondary points in Europe. The incoming transatlantic was late and the continuing flight number, often the only flight of the day, had left.

You can still find the equivalent happening in the US at major hubs nowadays.