I think you just get damage bonus from skills if your girls are equiped with a 6* equipement (personnal or event one) with world flower typing matching their own nation and enemies nation weakness. I almost never bothered with that stuff and just brute force through stages the good old way. Even on Nation missions based on this mechanic. I don't even know how this stuff actually works.
The bonus from 6* personal equipment seems way better than the nation typing things anyway.