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Tear Jerker: Ōban Star-Racers
Rick being told he can't race anymore was quite upsetting, especially with the tear that rolls down his face while watching over the stadium.
The entire subplot regarding Maya's death.
In "Monstrous Like Muir", we learn that Muir wishes to revive his mate, who was killed by alien hunters. Muir had hoped winning the Great Race would give him the power to bring her back to life. Sadly he, like Molly learns that the Avatar doesn't have the power to resurrect the dead. While he can't genuflect like a human, it wasn't hard to tell he was crying on the inside.
While Molly/Eva's Calling the Old Man Out moment with her father was also a Crowning Moment Of Awesome, it also makes you sad when watching the scene. Eva lets out all the pain she felt during her time with her father during the racing games and her how the entire childhood she has had without her father.
Don Wei's My God, What Have I Done? reaction when Eva/Molly recalls how he didn't recognize her after they met in the first episode is also heartaching. You can see on his face and hear in his voice how much he regrets it.
The end of the series when Jordan becomes the Avatar in Molly's place after confessing his love to her, and then having his last appearance of the series (outside of flashbacks) being him shedding a tear as he watches Molly leave Oban.