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Tear Jerker: Forward
Forward has a number of these, mostly centering around River, as the author makes a point to run her through the wringer at least once per "episode." They include several violent mental breakdowns on River's part, as well as River and Jayne being tortured by Niska, and one bit at the end of the "Mosaic" episode where River has an Out, Damned Spot! moment while trying to wash off the blood of all the men she'd killed the previous chapter.
One reviewer pointed out a particularly potent moment on River's part toward the end of "Business," when she starts shooting at the Hands of Blue that are after her.
River: "You want her back? This is what you want? This is what you made, and you want it back? Not yours! Never yours, won't be again!"
Another reviewer admitted she started crying at the end of "Condor" when Mal talks to River about how he doesn't know when he finally decided she was part of his crew, but that at some point he decided that fighting for her, like fighting for the people of Miranda, was a cause worth dying for.
The ending of the fifth chapter of the "Adrift" arc, where River is fighting the Reaver horde and is hit by paralyzer darts. It ends with the "Riverthink" suddenly failing, and the last words of the chapter being River desperately apologizing to Simon as she falls.
What makes it even more potent is that the author normally ends a chapter with a few paragraphs of commentary. That chapter? Only his Catch Phrase.
River gets another gutwrenching moment in the epilogue to the "Silver" arc, which features her tearfully confessing to Book that she went after one of the men who tortured her at the Academy, but who himself was deeply traumatized by what he put her through and was left an emotional wreck afterward. The Tear Jerker comes while she is flashing back to the confrontation, where she has the man at knifepoint, and realizes that she doesn't hate him, when she knows she should, and she keeps slamming him against the wall and screaming that she should and demanding to know why she doesn't want to kill him. He tells River it is because he doesn't believe she is a murderer, and she then has a complete emotional collapse as she realizes he isn't an evil person and that she has no idea what to do or what to think.
The knife came away from his neck and clattered to the floor, and the sound shattered the spell. The dam broke, and the girl fell forward, sobbing, head pressing into his shoulder.
Rishard Dannet did the only thing he could. His arms rose up and wrapped around the girl he'd helped break, and he held her tight as she collapsed, whispering in her ear that he was sorry for it all, that she was okay, and that everything was okay.
In "Hunt," Zoe has a single gutwrenching line, mostly because in the previous arc she'd been developing an intense, maternal protectiveness of one of the child refugees at the abbey, only to discover said child was a psychic Inducer who was influencing everyone, including Zoe, and actually a psychopathic murderer. She sums it up thusly:
Zoe: " She got in my head."
River's letter to Simon at the end of "Hunt." She tries to apologize for what she did and tells him how afraid she is of her abilities and how she was being used, and promises him that she will get better, but that she has to do it on her own. At the end of reading it, Simon breaks down and starts crying.
The very last scene of "Hunt" has Wash trying to comfort Zoe as she breaks down crying in the aftermath of having to kill Kathryn.
Once again, from "Hunt": As Katie lies dying after Zoe shoots her, she finally has a moment of clarity where she realizes that what she's done was wrong.
"It was wrong," Kathryn said, and Zoe saw tears forming in her eyes. "It was wrong, wasn't it?"
"Yes," Zoe murmured. "It was."
"I'm sorry," Kathryn said. It came out as a hoarse whisper. "I'm sorry I hurt . . . everyone. Tell . . . ."
The Fourth Interlude, when we see John and River's first meeting, and he asks her why she held out for so long before trying to kill herself. Her reply is heartbreaking: "I am alone. Deep down, beneath the ground, where I will never make a sound."
In "Last Man", the death of the American, who dies saving Jayne. Even worse, he's strongly implied to be Jayne's uncle.