Time Rifts, who they're tied to and what they represent. To me at least (SPOILERS!!!)

[794redfield]

Again, just to clarify, for those who have NOT played this game, this will spoil pretty much EVERY Purple Time Rift. If you want to keep them a mystery, do not read this, thank you.

Mafia of Cooks.
Person Hit: Mafia Goon.
Rift Reflection: As the Goon at the very beginning of the Rift says, Time Pieces that break as a result of Hitting somebody create a Rift formed from their inner thoughts, likely being influenced by the victim's mental state at the time of being struck. The Mafia as a whole are rather "Simple" men, as a result, the Time Rift is very basic in its layout. Nothing else apart from Mustache Girl messing with them has ever really happened on the Island since, so what we see is what we get.
Pictures' Meaning: The Goon that was struck was reminiscing on what lead the Mafia to arrive on the island they would turn into Mafia Town. The Mafia as a whole ran their own factory, but their Boss was feeling either down or just bored from the constant work and decided to take his men out on a voyage to find a place where they could just relax and do their own thing, which, was honestly what they were doing on the entire trip at sea. Eventually they came across a nice little Island and quickly claimed it as their own, and the rest was history.

Dead Bird Studio.
Person Hit: Either the Conductor or DJ Grooves (Likely the one whom you face as the Boss of the chapter as this Rift being created would also cause them to learn about the power of the Time Pieces)
Rift Reflection: Regardless of which Director got struck, they both have the same goal, to one up their Rival and win the Annual Bird Movie Award every year (For some reason this has been going on for at LEAST 42 Years and neither the directors, their workers and even the Studio staff seem to have aged a day beyond young adulthood) to this end they're constantly running through memories of the past to try to improve their craft each year in the hopes of winning. However... no matter which of the 2 got struck, both have a nagging suspicion that their Rival has been cheating in one way or another, causing the Rift to seem broken and in disarray, reflecting that their paranoia at the thought of being cheated has caused them to become overly obsessed with either perfecting their current movie or revealing their Rival's deception by any means possible.
Pictures' Meaning: This is why it could be either Director, since it shows the ambition of Both the Conductor and Grooves from a young age, one living in the vast deserts and canyons, and the other living on the moon, both wishing to become renowned Movie Directors in the hopes of winning Annual Bird Movie Awards. Years later when they become Adults, they each buy a Key for Dead Bird Studios, which they assumed would be their's alone, once they realize they both bought and now work at the same place, they became Rivals from that point forward and have been competing with each other for the Awards ever since.

Sleepy Subcon.
Person Hit: We don't really know honestly, frankly, it couldn't have been Vanessa since she wouldn't have known that the Prince was really buying flowers. Nor could it have been Snatcher, since if he knew what the Time Pieces could do, he'd NEVER let Hat Kid take them back. My best Theory is it was a former Servant to Vanessa, since anybody close to Her would very likely know the real story.
Rift Reflection: This is a bit of a harder one to explain, since we don't know anything about the person that got hit. My best theory is the Rift reflects their inner thoughts of "What lead to our Queen going mad?" when Vanessa and the Prince were in a stable relationship, she was happy and open with her subjects, reflected by the first 2 parts of the Rift being wide open, but after she saw the Prince interacting with a Flower Girl and assumed he was cheating on her, she started closing herself off from the rest of the world, becoming more and more cold and distant until she eventually turned on her Subjects and Servants alike, represented by the later maps getting smaller and smaller with the last one before the Time Piece being a tiny enclosed tower.
Pictures' Meaning: These likely represent the Servant finally piecing together how everything went wrong, their Queen and the Prince were deeply in love and it made Vanessa a joy to be around, but one day she witnessed her Prince with another woman, assuming she was being cheated on she fled back to her manor in tears, but the Prince was actually buying flowers for her, which he would very likely relay to the Servant when going to see her. But Vanessa still suspected he had scorned their relationship and refused to hear him out, instead having her Guards lock him up in the dungeon where he would rot away as Vanessa descended completely into madness and became the monster we know her as now. With their Queen having gone completely mad, the Servant either fled the Manor to parts unknown, but couldn't bring themselves to leave the Forest entirely or became one of the many Ice Statues within the halls of the Mad Queen's residence.

Alpine Skyline.
Person Hit: Alpine Goat.
Rift Reflection: The goats don't really pay any mind to anything or anyone around them, and if something does get in their way, they just push it aside. To put it bluntly, their heads are in the clouds, constantly thinking about the world around them and rarely taking notice of others. As a result, the Rift has very few enemies to speak of, and the ones that are there are annoyances at worst (Crows because they barely qualify as a threat and the Cats because they'll steal from anybody they see, usually to their frustration), the rift is more about visual spectacle and exploring the heights like the goats themselves think and act.
Pictures' meaning: This basically reveals what the Goats hope to obtain, Spiritual enlightenment, by cutting themselves off from mortal distractions and becoming one with the world around them, one of their own managed to do this and enter the Twilight Realm as an enlightened being and now all the other Goats are trying to follow in his footsteps.

Deep Sea.
Person Hit: Captain Walrus.
Rift Reflection: Despite his rather "Indifferent" exterior, the Captain has always had ambitions of being a highly successful sailor, but ever since a tragedy that happened a while back, he sometimes begins to reminiscence about past dreams and what could have been. Exploring the Deep Sea with the person that he had admired, with the final map before the time Piece symbolizing his dream falling apart once he learned of the tragedy the Pictures show.
Pictures' Meaning: These reveal the reason for the Captain's cold and stern outer personality, back then, he was taught by a Seal mentor in the ways of sailing and running a ship, causing him to admire his teacher and possibly run a ship along side him in the future. When his chance came, he signed on to a Ship that would have Seal crew mates, he had assumed this meant he could get to sail with his mentor, but he realized too late that they were manning a different ship. This didn't discourage the Captain too much however and he continued to daydream about sailing alongside his teacher, that is... until one of his crew brings him a letter of his Teacher's demise, going down with their ship. This news effectively crushed the Captain's dream and caused him to become both a frequent smoker and act generally cold around others.

Tour.
Person Hit: Mustache Girl (In fact... I think you HAVE to choose to throw a Time Piece down to her in order for this Rift to appear.)
Rift Reflection: Needless to say... at the time Mustache Girl got hit, her mental state was in a lot of turmoil, literally minutes before, the entire population of the planet was screaming at her to get lost and she had been beaten unconscious by the one person she believed she could have been best friends with if they hadn't been "selfish", choosing to keep the Time Pieces for themself rather then use them to help others and after all was said and done and Time was restored, she began flashing back to everywhere she had been trying to figure out how everything fell apart for her. While also realizing that what she did at Time's End whether she accepts responsibility or not, was just a bad idea in general and a grave mistake she'd rather forget, since its the only part of the Rift that's in a state of disrepair, while all of the other areas are intact, though muddled, her castle at Time's End is completely in ruins.
Pictures' Meaning: What these represent is likely Mustache Girl coming full circle and remembering what her original goal was in the first place, to run the Mafia off her home for ruining it. She remembered the time she spent with an old man who is either her grandfather, or just an elderly resident when the Mafia had docked at the island, at first she greeted them happily, but in response, they pushed her aside, began mugging the residents and were generally ruining the place, turning a once beautiful island into a smoggy shell of its former self. Outraged, she tried to fight back, but got roughed up for her efforts and stole a Medallion from Mafia HQ while also swearing to their boss she would run them off the Island, much to the Mafia Boss's amusement, she somehow got a Red Hood as well and began doing whatever she could to generally mess with the Mafia while taking residence in a small alcove on the beach side.

[JDWMGB]

GL with updating this for Rumbi's Fun Factory

[The Void Boy]

I'm pretty sure the tour rift can be played regardless of the choice you made. But I would agree that throwing the time piece to Moustache Girl would be the cannon reason why the time rift happens.