Hi there!
To clarify, in each Lodestar, every player takes turns being the focus when searching for their memento, while the other remaining players would help them in their search!
Just like in a couple of GMless, No dice no masters or Belonging outside Belonging games, scene framing and roleplaying is rather freeform and very much a collaborative conversation between players on how to frame a scene/narrative, where to go next, and how to give voice to certain people that you come along your way. Any player can pick up the mantle of describing a scene and or voicing an NPC if they so wish, including the spotlight character.
That being said, when it comes to the beginning of searching for the spotlighted player's memento, they should establish the vague sense of where/what their memento could be like, and give suggestions on where the table could begin their search.
In other words, the spotlighted player should suggest an initial frame at the start in accordance to the 'vibe' of their memento and the table can play it off from there. Entirely up to you if you'd like to have each player narrate a scene in table order, or have a more freeform roleplaying experience from there!