Moves that buff/debuff, ie. Psychic with Sp.Def drops, Ancient Power with all stat increases, will apply the buff/debuffs on the same turn as the buffs/debuffs were given.
Examples: A pokemon using ancient power and getting the boost will do 1.5x damage on the turn the boost was given, rather than getting the buff after the moves damage was applied and the boost being available next turn.
A pokemon using psychic and getting the applied Sp.Def drop will do 1.5x damage on the turn the debuff was given, rather than getting the debuff after the moves damage was applied and the boost being available next turn.
What I've also noticed is that it seems inconsistent between different moves, for example Superpower will do the correct amount of damage respective to the amount of attack and defense drops it has received over the battle.