Water analogy still applies for that. If the pipes are already full of water if you inject water on one end it almost immediately comes out of the other, even if the water is only being injected at a slow speed. It’s not immediate because the moment you inject water the water in the pipes first compress and that compression travel like a wave much faster than the water.

The moment the water analogy falls apart is when you depend on magnetism, because electrons moving cause a magnetic field around, but water moving doesn’t generate any field around it, so it can’t imitate an electromagnet or a transformer.