Short answer: No, you will always have a party available at the correct level when you need to clear a dungeon for story progress.
While you are going through the MSQ, your trusts will be in "scenario mode". You'll have a different group of options for trusts according to who is with you at that point in the story.
After clearing the dungeon but before reaching the end credits, you can replay the dungeon in scenario mode.
Once you clear the credits, you no longer have access to scenario mode and instead you're given "avatars" of your final party group who you will need to level up. They can run any dungeon regardless of whether the character was originally present or not, but you need to get them up to the right level for the dungeon first.
After getting all six to level 80 you can access scenario mode again if you prefer.
Also for post-credits dungeons you can initially access the scenario version while you are progressing through the MSQ, but after that you need level 80 avatars if you want to run it again.
