Right off the bat, I should clarify that there is literally zero canonical evidence for this theory. This is solely a fun idea I have based on the lore of the Legend of Zelda franchise and the Four Swords manga itself. This theory is somewhat tied to my "Brain Fuckery" headcanon, but I will explain that in detail on the page for that headcanon.
For this theory to make sense, I will explain how I think the Four Sword is able to create copies of the wielder and how this has changed over time with subsequent Four Swords games. Canonically, the Four Sword originally existed as the Picori Blade and was gifted to the Hero of Man. In the time of The Legend of Zelda: The Minish Cap, the Picori Blade is shattered and reforged as the White Sword. In the same game, Link gathers the four Elements to convert the White Sword into the Four Sword. Every time he infuses the White Sword with an Element, it receives a power-up. Specifically, after the second Element is infused, the sword gains the ability to create one copy of Link. For the third and fourth Elements, it adds one more copy each for a total of three copies.
The ability of the sword to create copies of the wielder, then, seems to rely on being infused with Elemental power. It can perhaps be said that each copy is powered by a specific Element. These copies imitate Link exactly and look no different from him except that they are made of light rather than organic matter, causing them to dissipate if they walk into a wall and also causing them to vanish after a short amount of time. Link must also charge up his sword while standing on special tiles to manifest the copies.
In the next Four Swords game, The Legend of Zelda: Four Swords, the copies created from the Four Sword appear as soon as the sword is drawn. They are corporeal, and they can move independently of each other. Three of them also wear a color other than green.
In the final Four Swords game, The Legend of Zelda: Four Swords Adventures, the copies function much like they do in The Legend of Zelda: Four Swords, with the only difference being that their voices are different pitches of Link's voice.
Without going on a long tangent, I believe that the Elemental power inside the Four Sword decreases over time for a variety of reasons, necessitating that Link collect Force Gems in the final game to restore the Four Sword to its true power - something that was not needed in either of the previous games. (By the way, if anyone wants the long tangent version of this page, please email me!)
The Four Sword's power is not solely to create copies of its wielder and dispel darkness. It also has the ability to lift curses from people during the era of The Legend of Zelda: The Minish Cap, and it has the power to reverse time during the era of The Legend of Zelda: Four Swords Adventures - specifically when it lacks sufficient Force!
Because the Four Sword's ability to create copies of the wielder seems to be powered by the Elements, it is not unreasonable to say that perhaps those copies are manifestations of those Elemental powers - and perhaps those manifestations have some ability to manipulate the elements in turn. This means that you can totally make a magical girl AU for the manga where each Link controls a certain element. Where am I going with this? Well, consider that Green is made from wind, Red is made from fire, Blue is made from water, and Vio is made from earth, and they have some influence over their respective elements. Have you considered how the winds in the Desert of Doubt stopped blowing after Green made it out of the Zuna Pyramid alive? Have you considered how the river ferrying the Links away from the Village of the Blue Maiden became rough when Blue became angry at Green? Or how about Death Mountain erupting after Shadow Link was killed by the Links the first time? These incidents might be reflective of certain Links' emotional states.
(By the way, here's my favorite headcanon related to this theory: Remember how the plaque with the Hylian crest falls off the wall/ceiling onto the floor near Green inside the massage tent, waking him up and making him remember his quest? This happens right after Vio is taken away by Shadow in the Lost Woods. Why did the plaque fall? Well, perhaps Vio, unaware that he can even manipulate earth, made the Earth quake so that Green would awaken. He wants to be rescued! He needs the others to find him.)
Now, consider that Force is defined in canon as the energy that flows through all things. Surely, if this Force flows through all things, and the Links have some innate ability to manipulate those things through Elemental power, it isn't a stretch to say they may also be able to manipulate things because they can use Force. As we see in canon, they are able to translate Force into light beams and seem to draw strength from it. They fight each other for the chance to gather Force, almost as if fighting for food, and when Green is sapped of Force by the monsters in the Zuna Pyramind, he seems to become discouraged and upset. Even when the Force is shaken out of him by the Hinox and then stolen by Tingle, he seems especially upset. This all indicates, at least to me, that perhaps the Links draw their power - to exist, to fight, to manipulate things around them - from both their Elemental powers and Force. Let's take this a step further: Force is in all things, including things like tornados, wildfires, waterfalls, and volcanos. Couldn't it mean that the Links, also being things that exist in the world and having a unique, innate ability to utilize Force, are essentially in tune with the universe? And in that case, couldn't they revert the universe to a previous state if needed? Like, say, traveling back in time using the power of the Four Sword?
This theory is easily disproven, I'm sure. But I really like the idea that the Links are beings made of Elemental power and Force that are in tune with the universe and can manipulate their respective elements. For time manipulation specifically, it should be noted that all Four Swords are needed to do this in The Legend of Zelda: Four Swords Adventures, but this makes sense when you think about it. Need to revert everything to an earlier state? You need to cover all elements of the world to do that - and wouldn't you know it, we have control of all four of them! This theory then also explains why Force is so precious to them, why it affects their mood and being, and why they can use it in ways no one else can.
Now, if you'll let me offer one more headcanon... You know how Red and Blue could hear Green calling for them from the Desert of Doubt despite not being anywhere near him and hours after Green had been calling for them? Perhaps, if the Links can manipulate time to some degree, and they are able to manipulate the space around them to some degree, they may also able to reach out to each other across time and space. They are like beacons to each other in the vast void of the universe, and I firmly believe this is because they are linked by their shared soul, their Elemental powers, and Force. Notice how the Four Swords all resonated with each other once they got the last bit of Force needed to restore their power. They are tied to each other no matter what. They need each other. They want to be with each other. How beautiful is that? The universe calling out to itself, only to hear its own voice and feel some sort of comfort. It's not a coincidence, in my mind, that the resonance gives Green strict reassurance that the others are okay and motivates him further to search for them. Love is at the core of the universe, and love is power. Amazing.