During the Sixth Generation, the Pokemon Company released Pokemon Omega Ruby & Pokemon Alpha Sapphire after the games of Pokemon X & Pokemon Y. These games brought along many new Forms and items that weren't previously coded within Pokemon X & Y, thus causing compatibility issues between these four games. And now, with the new content found only in Pokemon Ultra Sun & Ultra Moon compared to Pokemon Sun & Pokemon Moon, the same issues are arising once more.
New Pokemon and Forms found in Pokemon Ultra Sun & Ultra Moon - like the four New Ultra Beasts and three different forms of Necrozma - cannot be traded to or battled with Pokemon from Pokemon Sun & Moon. Pokemon with brand New Moves (including ones taught from Move Tutors) and Items introduced in Pokemon Ultra Sun & Ultra Moon also cannot be traded to or battled with Pokemon found in Pokemon Sun & Moon.
Basically, if it something cannot be found in Pokemon Sun & Moon, you cannot trade it or battle with it from Pokemon Ultra Sun & Ultra Moon with said games. The only things found in Pokemon Ultra Sun & Ultra Moon that can indeed be be given or used against Pokemon Sun & Moon are that of Totem Sized Pokemon; however, only Totem Sized Pokemon that were originally found in Pokemon Sun & Moon as Totem Pokemon can be done so with.