Even before Dave started to live on the meteor, Karkat has had troubles trying to not break down in front of everyone. It was considerable harder for Dave to try to talk with Karkat, considering how sad he is inside. Dave gave him some time, spending most of the time of the first year with Terezi.
He learned though that his original thought and actions taken because of that thought caused Karkat more harm, so Dave started to spend some more time with the Vantas. Over the next year, Dave gradually spent more time with Karkat and less time with Terezi.
When the third and final year was being traveled through, Dave slowly began to see Karkat's soft interior. Unlike other people who would just say that Karkat isn't being himself, Dave listened to the Vantas, helping him with his troubles.
With the growing trust between the two, Dave allowed the troll to see his own interior of helplessness and doubt. After time spent between the two in privacy where just about their troubles, Dave wondered if trolls called their best friend something different. One day, Dave asked about it, and the Vantas told him that their term for "best friend" is moirail.
That day, Dave actually listened to Karkat rant about the quadrants, finding connections to human terms with them. Soon, Dave knew the troll quadrants as following: Matesprit is boyfriend/girlfriend, Moirail is best friend that you trust fully, Kismesis is the person hate yet is also hot to you, and didn't have a clue for the last quadrant that he couldn't even remember the name to.
Karkat was willing to try to explain the Auspictis (My apologizes if I spelled that wrong. I'm still new to that quadrant) quadrant to Dave, but declined kindly, not wanting Karkat to go over-board with the troll love.
A few days after that, Kanaya and Rose started to tease Dave about spending so much time with Karkat. Using his new knowledge on the quadrants, Dave said that Karkat and he are most likely moirails, still not completely term himself. Kanaya was interested in why he said that, and Strider tried to explain why he said it.
It was true though. Karkat and Dave were now technically moirails. They fit all the requirements to be moirails, but the true test was just beginning. Moirails need each other to support each other. Yes, Karkat and Dave did help each other with their internal turmoil, but it was yet to be seen if they actually help each other be better people in life.
It seemed that they did need each other, as the end of the third year came, and they were almost at the new session. Dave and Karkat were almost inseparable, and they helped each other understand things about the two species both of them are in, and other people's actions. They guided each other to better decisions, which sadly makes this the doomed timeline. The Alpha time wasn't supposed to have Dave and Karkat moirails, but that never crossed the two males' minds.