In addition to the timed events on given days after starting the Special Demo, there are also a number of special occurrences that place at given times. Again, these don't add much value to the Demo, but yet are still enticing lures for the upcoming games.
Hau: On Sundays when it is at night in the Demo, Hau can be found in the Masalada Restaurant instead of the Pokemon Center.
Mother: On Tuesdays, you will find your mom exiting out of one of the Shops in the city.
Lillie: On Fridays you will find Lillie looking out at the sea across from the Pokémon Center. She will be whispering to her Pokémon in her bag, saying that she wishes she could share the view with them. Once she notices you, she will walk off.
Beach Girl & Alolan Dugtrio: On some days in Ten Carat Hill, you will find a girl on the beach. If you Talk to her, she will enhance your PokeFinder so you can take a picture of the Alolan Dugtrio on a nearby island.
Dexio & Sina: One some days, you will find Professor Sycamore's (Kalos Region Professor) Assistants Dexio & Sina in the Pokémon Center. They will tell you how they are on vacation.