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Tear Jerker: The Expendables
Billy's death, that is so excruciating and made quite one of the things that became personal for the team.
Tool's monologue in the original and Billy's in the second, when they both reveal what made them the shell shocked veterans they are now.
There's a certain recognition of the passage of time at the very end of the sequel, when in an extremely meta moment, Barney comments on his new plane, "That belongs in a museum", to which Trench replies "We all do." A recognition that their age is coming towards an end.
In the third, Galgo's actual backstory is this. His former team was in a rescue mission similar to the one he and the Expendables are in, all went bad, and he was the only survivor.
Still in the third, Caesar being shot and the events that ensue, leading Barney to fire his team so that they will live. The latter part doubles as a moment of Heartwarming.