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Tear Jerker: Air Gear
The deaths of Spitfire and Aeon Clock. Afterwards, Kazu gets picked up by Kaito and Akira and all he can do is sob in the back seat, holding onto the Flame Regalia because they're all that he has left of Spitfire. Not even Kaito wants to take them from him. Ine Makigami, who is essentially Spitfire's lover, is shattered by the news.
On a similar vein we have the death of Yoshitsune. We don't even know it happened until Benkei delivers the Rumble Regalia to Kogarasumaru. She's missing a leg. That just makes it worse, not because she's down a leg, but because she really doesn't care. It's nothing compared to what happened. Trident was wiped out and her friend and leader was killed. After delivering the Regalia, Benkei finally asks Yoshitsune if she's allowed to cry.
Nike's backstory is one as well. While he's definitely a sociopath, it's because he was used his entire life by his brother while in his own mind, the two are closer than anything, making him excessively loyal. After Nike is defeated, Sora's casual dismissal of Nike as a useless idiot really demonstrates how deluded Nike is and how pointless his life was.
Gazelle's death. She and Kaito ran so far, fighting a hopeless battle all the way. When the Takeuchi Brothers finally catch up with them to take the Thorn Regalia she wore and the code in her mind, Kaito blows up their only weapon in the hopes of taking the brothers down. That, of course, fails, and realizing that there is no way to beat the brothers, Gazelle takes Kaito's gun and kills herself, depriving them of at least the code they need to activate the Sky Regalia, while Kaito helplessly screams for her to stop.
A minor one. Ringo's always wanted nothing more than to help and support Ikki but has never had the chance to do so, at least openly, because of the laws of the Trophaeum and of Sleeping Forest. Now, during the invasion of Genesis' Aircraft Carrier, when Ringo's finally able to do so, the Storm Regalia are irreparably damaged because Kururu shielded her when she got a bit careless. The utter helplessness that Ringo feels as Kururu tries and fails to repair the Regalia is just heartbreaking.