Spoilers Off applies to all "Moments" pages, so all spoilers are unmarked!
If Part 4 Jotaro and Kakyoin are fighting against each other, Jotaro says to him "Kakyoin, Long time no see", giving us a Call-Back to their first meeting. If Kakyoin defeats Part 4 Jotaro, he asks him what he's like as an adult. Jotaro's winning quotes however.
Jotaro (Part 4): "It's good to see you again."
Jotaro (Part 4): "I won’t forget you… from now on, either."
This after-battle dialogue between Part 4 Jotaro and Avdol:
Jotaro (Part 4): "I turned 28 this year, Avdol."
Avdol: "Ten years have passed? Time sure flies."
Similar to the pair above, Old Joseph and Caesar have special dialogue as well. Caesar being oblivious the person he's fighting with or against is actually Joseph, and does not find out until after the match. Joseph, on the other hand, recognizes Caesar immediately— despite previously not realizing who Part 1 Speedwagon nor his own younger self were.
Old Joseph: "Is that you, Caesar? Is that really you?"
Jolyne screaming, "It's all your fault I'm like this! YOUR FAULT!" at Part 4 Jotaro. Ermes also asks Part 3 Jotaro to be more attentive once his daughter is born.
The sheer shock and disappointment in Part 8 Josuke's voice after meeting Part 4 Yoshikage Kira and being told he's a serial killer.
Josuke (Part 8): "Wait! Are you really Yoshikage Kira?"
Kira: "Indeed...but not the one you're looking for."
Old Joseph stepping in to prevent an argument between his younger self and Caesar, specifically, the one that led to Caesar running off and getting himself killed. He lectures the younger Joseph on not being so cocky and thinking he knows everything, while he warns Caesar that while his desire to avenge his grandfather's death is noble, he shouldn't forget about those who care about him and are still alive... His death would probably cause a lot of pain and guilt for them.
The fact that Joseph also knew exactly what to say to de-fuse the situation too. He’d been replaying the same conversation over and over again in his head out of regret throughout the years.
If Josuke gets KO'd first with Old Joseph as his partner, Joseph laments how he should've been a better father to him. If Old Joseph gets KO’d, Josuke yells at him not to make his mom sad.
Overlapping with Harsher in Hindsight: Josuke getting to meet Part 2 and 3 Joseph, who are both a far cry from the senile old man his father becomes in Part 4.
The only character that audibly cares that Dio Brando is KO'd? Jonathan. He still can't bear to see the man that he hoped to call his brother one day in pain.
If Caesar and Young Joseph are on a team and Caesar is KO'd first, it replays Caesar's last action of trapping his bandana in a blood bubble and sending it to Joseph, who enters Soul Succession with a unique animation where he grabs it and screams Caesar's name, much like his death in Battle Tendency. Caesar even screams "Jojo!!!" if Young Joseph dies first, although there's no unique animation.
Dummied Out dialogue for Soul Succession has Old Joseph call out Caesar's name exactly as he did fifty years ago, once again forced to watch his friend perish and still unable to do anything to stop it. Conversely, if Old Joseph dies instead, Caesar will scream his name the same way Joseph did his in canon.
Caesar: "JO-JOJOOOOOOOOOOO!!!"
If Johnny is partnered with Gyro and KO'd, Gyro's animation and reaction will be yet another Call-Back to Caesar's death, only this time the Zeppeli will be mourning the Joestar.
For all of his other underlings, even his past self, DIO's Dummied Out reaction to them being KO'd is of disinterest and contempt at their weakness. When Pucci is KO'd, DIO somberly realizes that he now has to spend his long, long life having outlived the one man he trusted his life to.
The Dummied Out version is even worse, as he says "Am I...really going to lose you here...?" in an equally somber tone. Carrying heavy implications that DIO had already come to terms with the likely possibility of him longoutliving Pucci, but now he's mournfully processing how that moment has now suddenly arrived, rather than happening later.
to prevent an argument between his younger self and Caesar, specifically, the one that led to Caesar running off and getting himself killed. He lectures the younger Joseph on not being so cocky and thinking he knows everything, while he warns Caesar that while his desire to avenge his grandfather's death is noble, he shouldn't forget about those who care about him and are still alive... His death would probably cause a lot of pain and guilt for them.