Successfully complete the indicated task in the demo version of Pokemon Sun/Moon to unlock the corresponding item in the full version of the game using the "Pokemon Link" option:

Ash-Greninja (Greninja with Battle Bond used in demo): Successfully complete an adventure.
Balm Mushroom: Celebrate the birthday of the Pikachu in the corner of the City Hall 12 days after starting the demo.
Comet Shard: See the girl in the Marina off on her journey 24 days after the demo starts.
Nugget: Successfully complete the Pokemon Capture challenge.
Pretty Wing x10: Speak with the man outside of the Pokemon Center one day after starting the demo.
Star Piece: Defeat all of the trainers at Ten-Carat Hill.
Stardust x3: Get rid of all the rocks in Mahalo Trail.