The quiet moment with the combined pair of superteams, in which there's reconciliation, reflection, and in a tearjerking moment, Hal Jordan and Barry Allen (both deceased at the time of the series) talk about missing the opportunity to have a picnic with their wives together, topped off with a talk between Superman and Captain America in which they reveal their private fears; Superman worries the JLA is doing too much for mankind, stifling its growth, while Cap worries that the Avengers aren't doing enough. They acknowledge just before returning to their own respective dimensions that it doesn't matter in the end, so long as they fight the good fight.
Cap loaning his shield to Superman.
Superman: I'm...I'm honored, Captain. I'll do my best to wield it well.
Green Arrow, thinking Hawkeye has been killed, takes his arrows and vows to make each one count. Throughout the series, their massive egos had clashed over who was the better archer.
Speaking of which, Hawkeye is also now the only Marvel character to have been officially inducted into the Justice League of America.
During the final battle with the various enemies gathered by Krona against the two joined teams. After Green Arrow apparently died in action, Wonder Woman ordered the rest of the heroes to press the attack, while she would take up the rear and fight off anything that might attack them from behind. As she prepared to face Surtur, her mother Hippolyta (the Golden Age Wonder Woman) and She-Hulk announce that they will fight at Diana's side. Shulkie's comment was especially in-character (doubling as CMoF):
She-Hulk: Yo, Star-shorts! Figured that you'd be getting bored, so I thought I'd hang with you. We can talk girl-talk. Y'know, butt-kicking, name-taking, like that.
Wonder Woman (Hippolyta): Exactly. And I will stand by your side too, daughter.
Wonder Woman (Diana): I... I'm honored... And very thankful... Onward, then... For Gaia's glory!
Even more Heartwarming in that Hippolyta was dead at the time, giving Wonder Woman one last time to be with her mother.
After the final battle, Superman discovers that he can't lift Thor's hammer anymore. Thor explains that Odin temporarily suspended the enchantment in order to let Superman wield it. But then he adds this:
Thor: Perhaps it was but briefly... but it was in good hands.
Barry Allen and Hal Jordan, commenting that the Flash and Green Lantern legacies are in good hands with Wally West and Kyle Rayner. This line from Barry is what really sells the deal (or it did, before the concept of Comic Book Death got so cheapened):
Maybe dying isn't so bad after all... Not if we leave a legacy... If we know someone will keep fighting for what we believe.
Vision giving up the last of his strength to help Superman fight Radioactive Man and Solaar.
When Superman mourns Vision later, Thor points out that he's been repaired after sustaining far worse damage before. He adds that Vision would have given more if he could.
Seeing Kismet and Eternity, the physical embodiment's of the DC and Marvel universes respectively, fall in love is a literal heartwarming moment between the two universes. Seeing them pulled apart is equally heartbreaking.
On a meta level, this symbolizes the respect and friendship that DC and Marvel hold for each other, regardless of their rivalries. After all, if the two big names in superhero comics didn't love each other, crossovers like this could never exist.
A minor one, but the gentle way Wonder Woman offers to prove the Grandmaster is telling the truth with her lasso - and she allows him to retain dignity while doing it by simply wrapping it loosely around his wrist three times.
Another minor moment, when they're making ready to leave the batcave. Captain America takes Batman's remark about Ben Grimm's "rough-edged charm" as an insult. Cap defends Ben by telling Batman that Ben's one of the finest men he knows.
As both worlds are on the verge of merging, Captain America snaps at the Justice League (mainly Green Lantern Hal Jordan). Superman gets angry and defends Hal from the Captain's accusations. True he goes a bit overboard, but his heart was in the right place.
Superman: You have no business criticizing a better man than you'll ever be!