Yeah, characters flip over to Eternal, but season-specific stuff won't.
From the site Q&A -
The character and progress on that character is transferred to the Eternal Realm. You can play this character whenever you want in the Eternal Realm. Specific Season-only features will no longer be accessible in the Eternal Realm. For example, at the end of Season 1, Malignant Hearts will disappear and be unslotted from jewelry at the end of the season. Infested sockets will become normal sockets.
Face the denizens of hell.
diablo4.blizzard.com
Some of that could change, but looks like they're setting expectations that nothing about the season mechanics will make it over.