Phantom Anon. Given the spoilers for each of the Pokemon Mystery Dungeon games, the only one one of the Links could likely take the role of the Player in is Gates to Infinity, so- each Link separately does Gates to Infinity, what happens? (I apologize if asking this ends up meaning I've put you in another research purgatory before you've even escaped the first one.)
First off - I assume that you've chosen Gates to Infinity because of the 'you can go back and forth' clause in the endgame? Which, yeah, that would have to be the case, Hyrule can't not have a Hero.
However, in the interest of sanity, I will not be rehashing the entire plot for every Link and will instead be giving snippets that I feel encompass the vibe of the individual adventure. Here we go!
Gen: a Snivy who will be learning every healing move available to him and will not be taking no for an answer. Also, Leaf Blade. "Stop recovering!" Kyurem snarls at him. "Make me," Gen snaps, and continues to spam way too many Giga Drains to be considered legal.
Speck: a Pikachu who technically isn't using Minimize, but also isn't not using Minimize. Nevermind that Pikachu can't learn Minimize, who told you that? He's clearly not doing it right now, see?
The Four: did you know that a group of Tepigs is called a sounder? And that a 2x weakness multiplied by four equals a really bad time for certain Ice-Type Dragons? And Ice-Types in general? And actually any Type at all when that much fire is involved?
Ocarina: an Axew, but there was an Incident somewhere along the way and now he's a Fraxure. This causes some very specific problems, mostly because nobody else seems to be able to evolve and how did he do that???
Mask: an Axew, but there was a different Incident somewhere along the way and now he's just. A Haxorus, sometimes. But only for about two minutes at a time, and also the Haxorus seems to be a completely different person? Hydreigon totally doesn't need to be concerned about the possibility that he may have accidentally stuffed two human souls into one body, nope, not at all.
Dusk: an Oshawott with really, concerningly high crit-rates when he uses his scalchop. Why is he calling his Slash attack a 'Mortal Draw' and why does it keep knocking out enemies in s single hit? That was a Steel Type, this is not how the type chart works-!
RGBV: a group of Pikachu is called a mischief and that's exactly what these four get up to. The Bittercold doesn't exactly have emotions, but if it did then oh boy would it be experiencing some frustration.
Realm: a Snivy who 1) has access to Grassy Terrain, 2) doesn't quite understand that Grassy Terrain is supposed to be an AOE move, 3) therefore somehow managed to interpret every piece of greenery ever as part of Grassy Terrain, and 4) has consequently and accidentally given himself access to the entire planet, because it's all in his Grassy Terrain so of course he can go there, that's only logical. He bypasses Magnagates simply by existing and breaks several previously-thought-fundamental laws of physics in the process. Espeon and Umbreon collectively lose their minds.
Lore: a Tepig who has been given unrestricted access to fire and nobody around to raise the alarm about it. His motto is "does it explode?" and if the answer is no then he sets about fixing it. This includes his own moveset. He treats Flamethrower like a fuse to a bigger explosion and it works. In unrelated news there's a weird Tepig advocating to add a new Type to the chart, he's calling it Kaboom.
Sketch: an Oshawott who makes a dedicated study out of not using a single Water-Type move. Is this wise? Probably not. Is it somehow working for him anyway? Surprisingly yes, denial is an excellent motivator. By the time Kyurem arrives on the scene he's divorced himself from the Wate-Type so thoroughly that he somehow counts as a Normal-Type instead.
Wind: did you know that Pikachu can learn Rain Dance? And on an unrelated note, there's been an extremely persistent and localized thunderstorm following Kyurem around for the past two weeks, he's looking a bit charred.
Steam: an Axew who casually claims he's being haunted. Nobody really believes him, until Hydreigon backs him up. "I did not mean to bring a second human soul this time," he says, staring at thin air. "Don't worry about it, she and I are a package duo," Steam says. "By the way, random question, but would you describe Kyurem as an 'empty husk' that may or may not be a functional open vessel? Asking for a friend."