REQUESTED BY: Mui_Mochi
A/N: This is assuming you're totally caught up on the whole manga and somehow get transported to the KNY world before the events of the latter half of the manga... AKA, Kyojuro and the others are still alive.
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Tanjiro:
He's very surprised when you manage to find the group and immediately go up to him with a big grin and talking a mile a minute, but once you manage to slow down and explain what you're doing there (and you promptly squeal and hug Nezuko), he decides to take you along (both to meet Oyakata-sama with your information and so that you can find a way back to your world).
Nezuko:
When you managed to find the group and promptly squealed when you saw Nezuko, she was initially cautious until you went to her, hugging her tight and gushing over her. She promptly blushed and, sensing no ill will from you, gave you a headpat. (You promptly resembled a tomato and squealed even more.)
Zenitsu:
Zenitsu was shocked when a pretty girl like you ran into him and immediately hugged him, squealing and fangirling over him. His entire face was red as you squished his cheeks and exclaimed how you never thought you'd ever meet him like this, and when you asked him to marry you he fainted.
Inosuke:
When you accidentally fell from the sky and landed on him, Inosuke was pissed. He demanded to know who you were, but once you started hugging him he grew confused. When you explained who you were and what was going to happen, he decided to keep you in the group, deeming you important enough to tag along (that, and you mentioned that you knew who his real mother was, and he was pretty curious about that).
Giyuu:
You had fallen from the sky and had immediately started talking a mile a minute, damn near hyperventilating when you realized just who had broken your fall. Giyuu was about to dismiss you as insane when you told him that you knew how to stop Muzan (and you knew what was going to happen to his comrades).
Kyojuro:
When you fell onto him from the sky and immediately gasped "You're alive?!", to say he was concerned was an understatement. He managed to get an explanation out of you, and while disturbed to hear about his future death at the hands of an Upper Moon, he determined that you needed to meet with Oyakata-sama to discuss how to take down the demon king.
Muichiro:
First, you fell from the sky, and then you were crying and hugging him, exclaiming that you were glad he was alive. Irritably, he asked you what you were talking about, and one long explanation later, he told you that you had to meet with Oyakata-sama to discuss it. He was very disturbed to hear about his impending death (and even more disturbed to find out that he was related to a demon, much less an Upper Moon).
Shinobu:
At first, Shinobu was trying to treat you for any injuries because you'd somehow landed in the backyard. But then when you started rambling about things that you claimed were going to happen in the future, she thought that you had a few screws loose until you mentioned how she was going to die. She listened, intrigued and shocked, as you detailed her plan to defeat Upper Moon Two, and at the end of it she determined that you needed to see Oyakata-sama to better discuss how to take Muzan down.
Gyomei:
First you fell into his arms while he was sitting beneath a tree, and then you were sobbing and telling him not to fight the Upper Moons. He managed to pry an explanation out of you, very concerned for your mental state, and when you explained who you were and how you knew he would die, he knew that you needed to meet with Oyakata-sama to prevent whatever deaths you could.
Sanemi:
At first Sanemi yelled at you for falling onto his head, but when you told him that you knew the future and how to defeat Muzan, he decided to give you a shot. It took you a while to explain everything, but once he learnt that Genya (and all but two of the Pillars) would die, he decided to take you to see Oyakata-sama. Hearing that he would die greatly disturbed him, and he needed to make sure that the man wouldn't meet the same end that you envisioned.
Akaza:
At first he paid you no mind, but when you mentioned the Flame Pillar dying at his hands, he grew shocked and decided to listen to your explanation. While he was pleased to learn that he would go toe-to-toe with the Flame Pillar, he was disappointed to learn that he didn't choose to turn. He decided to take you to Muzan in order to better plan for the future, as your viewpoint would come in handy (and he was intrigued with the knowledge that Giyuu, the Water Pillar, was such a worthy opponent).
Douma:
At first he thought you were crazy when you jumped up and started telling him that you could tell the future, but when you mentioned Kanae Kocho and her younger sister Shinobu (and the fact that Shinobu would be the one to ultimately kill him with poison), he became intrigued and decided to take you to Muzan.
Kokushibo:
He decided to leave you alone when he came across you lying in the forest, but when you realized who he was and blurted out that he would fall to the pillars, he paused. Something beckoned him to listen to your explanation of the future, and once you were done, he decided that you would be better off explaining it to Muzan.
Muzan:
At first, Muzan was just going to kill you when you saw him and started talking. But then you mentioned that he would battle with the slayers soon, and then he was forcing an explanation out of you. When you mentioned that he would ultimately lose, he became very bitter and threatened to kill you for even thinking it, but in the end decided to keep you as an informant on what actions the Slayers would take if his plans stayed constant. He was especially surprised to learn that Nezuko would conquer the sun soon (and that he would get even closer by possessing Tanjiro), and decided to put more effort into capturing the Kamados.
Yoriichi:
When you fell from the sky into his lap and promptly yelled about how you were in the past somehow, he grew concerned for you. But when you started talking about everything that would happen to him in the future, he was alarmed and saddened, both to hear that his brother took the path of a demon, and the fact that he wouldn't end up being the one to kill Muzan. He was happy to learn that his friend's descendant would finish the job, however.