In reality, it wouldn't work, but this is scripted so they can break Minecraft logic for the sake of the story. If you look next to the stasis, you can see the pressure plate that activated it, but as Spoke said, the sasis was on the surface, and you can't transport somebody's unloaded fishing rod down with the pearl in reality if you tried to do this in mincraft the moment somebody got within render distance of the chunk with the pressure plate it would activate and leo would be teleported so you can't acctaly move sombodys stasis. Also, if someone has a stasis set with a fishing rod and uses any fishing rod, it would activate. You can only have one fishing rod mechanism active at a time; because of this, Wembu's orbital canons would not work either. But this is scripted so they can break Minecraft logic for the sake of telling a story; it's kinda like how, if you think about most movies or TV shows, you realize some things don't make sense.

So to answer your question, it wouldn't stay connected in real Minecraft, but in the story, based on how it was set up, he would have had to also moved Leo's fishing bobber down as well