The two Mist Nin were disappointed that Kakashi killed Rin. We know next to nothing about her. Why were Rin and Kakashi so far afield from Konoha, without any obvious purpose?

Rin could have been an enemy agent. She was highly placed in Konoha (student of the Hokage) and could easily gain access to any number of secrets.

So here's the theory. Rin was a double-agent, either through forced means (kidnapping of family or the like) or through free will. She escapes Konoha with a secret (maybe even the extended information on the Hiraishin, personally written by Minato) and Kakashi realizes it. Maybe she's been acting off lately, maybe she's been asking weird questions. Maybe he just doesn't want to leave her side after Obito's death. He tracks her, catches her and forces her to reveal the reason why she left. Heartbroken, he is forced to kill her to protect Minato/the village as a whole. With her last breath, she conveys her regret over forcing him to do this, just as the Mist is about to reach them. Kakashi's Mangekyou unlocks from him killing his most precious person in the world.