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Tear Jerker: Temeraire
Any time a dragon or their captain is separated from one another:
Praecursoris being Put on a Bus close to the climax of His Majesty's Dragon, especially the fact that while he's being separated from Choiseul, his captain, the poor thing is screaming in anguish. It was so bad in-universe that Temeraire was an emotional wreck for days just from witnessing it.
During the portion of Victory of Eagles where Temeraire believes Laurence is dead, he curls up by himself at night and recites books from memory, trying to mimic Laurence's voice so he can pretend it isn't true.
Hell, the entire opening third of that book is one extended tear jerker.
Though there have been many deaths throughout the series, the death of Levitas in Temeraire was particularly heartbreaking. A thoroughly neglected and mistreated dragon, Levitas nonetheless showed the same compassion for his captain, no matter how abusive he was. Laurence, having managed to keep his calm with Levitas's captain so far, finally loses it when Levitas is dying of his wounds and Captain Rankin is drinking wine in the officers' club. He storms in, grabs Rankin, and forcefully drags him to Levitas's side. Rankin stutters a few reassuring words, although obviously forced, and Levitas dies with a smile on his face.
Laurence: "Levitas, come along now; look who's come."
Levitas: "My captain?"
Rankin: "Yes, I am here. You...have been very brave."
Levitas: "You came."
And shortly after, Laurence wraps it up with this jewel of a line:
Laurence:(shortly after Levitas dies, to Rankin) "Go. We who valued him will make the arrangements, not you."
Lien's reaction to Yongxing's death - the only person who cared for her and showed her any true regard. Laurence, remembering Praecursoris's reaction to being separated from Choiseul, is afraid she'll scream, but she's totally silent as she nudges at his body, picks detritus off it and then picks it up and flies away. Yongxing's death means that just about everything she does afterwards is for revenge.
Digby's death in Black Powder War, where he falls from Temeraire's back in a failed attempt to save a dragon egg which shatters in the process. Poor creature.
Everything about the dragons falling sick in Empire of Ivory. Everything from the slow, terrible deaths of the dragons, particularly of Obversaria and how badly Lenton's been affected, and even to the revelation that there's a possible cure. If you somehow didn't already see how much most aviators love their dragons, you'll see it here.
There's one scene towards the end of Victory of Eagles where Laurence and several others are attacking french camps, and Maximus leaps onto a french dragon to prevent her escaping, only to mistakenly break her back and kill her. As the attack continues, her captain rushes to her body and breaks down in tears; forcing his crew to drag him away when they are all finally captured.
The sinking of the Allegiance.
When Riley dies, Temeraire remembers how he was one of the only people brave enough to feed him when he was just hatched.
The loss of the Allegiance was only in part 1. Even by the time the book ends and much time has past since then Laurence hasn't gotten over it. And then in Blood of Tyrants Laurence loses his memory of the last eight years. He still thinks Riley is alive and has to be told otherwise.
The reveal in Blood of Tyrants that Temeraire's brother Chuan was murdered in a failed attempt to unseat Prince Mianning as Crown Prince.
Every moment in Blood of Tyrants involving an amnesiac Laurence and Temeraire before Laurence recovers. Dramatic Irony has never been so effective against the feels as you're stuck anticipating when certain events are brought up (or in part 1: when Temeraire and Laurence will reunite, but only physically). The strain amnesia, combined with Temeraire's naivety, puts on their relationship is felt, but it fails in comparison to the moment Laurence finds out he committed treason and has a Heroic BSOD. In response to this Temeraire has his own Heroic BSOD, believing that he's to blame for everything bad that has happened to Laurence. At this point, it looks like their relationship is never going to recover. Good thing that it eventually does.