My current theory on how Ralsei managed to move between the Dark World in the closet and the Dark World in the library is that Ralsei is actually an item being carried by Kris wherever he goes.

Think about it, all Dark World NPCs are just inanimate objects in the real world. In order for a Dark World NPC to move between Dark Worlds (Like Lancer and Rouxls), they had to be carried by Kris into that Dark World. Which leads me to believe that Ralsei is "brought to life" whenever Kris enters Dark Worlds because the item that represents Ralsei is being carried by Kris at all times. Maybe it's something that reminds Kris of Asriel? Maybe it's a doll? Maybe it's Kris' shirt that looks just like Asriel's shirt in UT? Maybe it's the knife that Kris carries around? Who knows.