I am running a game where I would prefer the characters personally feel invested in some NPCs and want to help them. I can near force the mission on them as "will of the gods," but I'd rather not.
The main mission is a rag-tag clan of ~20 very frail and disfigured outcasts are fleeing powerful, evil beings out to annihilate them. Frail and powerless now, but somehow very bad for evil later.
The PCs will get info [from the gods] saying the outcasts are super-lawful good; and it is imperative the outcasts get escorted to some destination to reveal their ultimate fate. This is sort of the "protect the president" guard duty for the PCs.
Right now, I have a setup adventure where the group will travel into the past and meet the group's ancestors. There, they learn cruel DNA type experiments are transforming them into these hideous creatures, changing their alignments, and they will continue to evolve. I'm hoping the PCs want to help or free them from the experiments.
The PCs return to the present, and at some point learn the descendants are in trouble and need help one more time, but it is very dangerous for the main mission.
How can I make them give a hoot and want to help, or even feel responsible? I'd rather not dangle treasure/magic items.
And, one issue is my current players have not stepped up to the plate in the past. They have let others/NPCs get killed rather than risk their necks when the situation looked, to them, as very risky. However, I may run this game again on a more heroic group, so the question remains the same.
Unlike normal NPCs, these characters will be gone for good when all is done.
Just to address some questions, this is not dependent on any game system although I'm running DnD 3.5 now. It could be 5e or other later.
re: Time Travel issues, I have contingencies if they don't free the tortured outcasts. Worst case is they don't help in the past, and later in current time they don't accept the mission. In which case, they learn later that another competing NPC team does it and gets fantastic rewards.
This "other NPC team" has already taken two other adventures they gave up on, and both times the players are pissed at their own failings, but then they do it again ...