What happens to companions that you finish one quest and then leave them for a while?

I thought it might be fun to do more rotational companions with in this playthrough. What happens if you finish a companion's quest and then leave them? Will the timer for the next one happen while not with me or brought back later? Will they still be waiting safely? Which ones can I do this with?
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Certain quests you should be able to do lateron. Some conditions get checked, e. g. Hexxat - if you did not recruit her and finish chapter 6, I think she vanishes. But if you do her first quest, and dump her, you can resume this lateron. I used to park them at the graveyard district; lateron when I used more mods, I would park them at the southern block or what the name was from Lava's mod (I can sleep there upon entering and it is a nicer place than all those undead nearby on the graveyard; would be nice if there were multiple large cities available).