NPCs disappear after completing side quest

I am interested in doing the side quests. To me it feels like I am helping the ordinary people in Eorzea and it also adds more story and background information on top of the MSQ. However, often after I finished the quests, the NPCs who give me the quest will run away or whatever, disappearing in the view. I wonder if I am helping the people or killing the people since they do not exist any more.

Can we keep them "alive" even they have fulfilled their duty to give out the quest? I feel somewhat strange and hesitant now when taking side quests.

Not all of them disappear. Some of the sidequests are given by the ones that are always there.