These series of quests will focus around the strengthening of individual skills, improving activation rates and/or effects. However, like everything else, there's specific conditions that have to be met, and apply:
before starting the quest, set your leader to the unit who has the skill in question. As an example, Protagonist would have to be the very first unit (leader) in your party.
afterwards, these quests can't be redone with the same individual unit.
however, different variants and rarities can be redone, because:
finally, these improvements only apply to the individual unit; the quest would have to be redone if you, for example (if Beastman ever did get a fix), want your Moritaka to also have the improvement you just did with Moritaka.
See Skill Evolution for more information about Skill Evolution.