This article really needs to mention Japan, who pioneered and perfected the idea and its smooth integration into the subway network. It's common in Tokyo for a train to start as a limited express that travels long distances at speed, then transition into an all-stops subway, tunnel through Tokyo, and then morph back into an express for the long haul out. Some trains on the Toyoko Line do this twice: express, subway i…
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Hannō > Kotake-Mukaihara: Seibu Ikebukuro Line
Kotake-Mukaihara > Shibuya: Tokyo Metro Fukutoshin Line
Shibuya > Yokohama: Tokyu Toyoko Line
Yokohama > Motomachi-Chūkagai: Minatomirai Line
Those are all different companies operating on the same rights of way seamlessly! When you reach the through service stops, it just looks like any normal stop - there is generally no interruption in service, it just stops for some seconds and proceeds onwards like nothing happened. If you visit Tokyo you would notice that sometimes you'll see Tokyo Metro-branded trains on JR tracks, or Toei-branded trains on Keio tracks, and so on.
Many lines are like this; if you look up any of the central railway lines in Tokyo on Wikipedia, you'll see on their station charts notes about their through services. Some more instances:
Odakyu Line > Tokyo Metro Chiyoda Line > JR Joban line
Keio Line > Keio New Line > Toei Shinjuku Line
Tokyo Metro Hibiya Line > Tobu Skytree Line > Tobu Nikko Line
JR Chuo Sobu Line > Tokyo Metro Tozai Line > Tōyō Rapid Railway Line, and so on.