In Linux the saves are in the home directory, with a subfolder path
../Larian Studios/Divinity Original Sin Enhanced Edition/PlayerProfiles/<ProfileName>/Savegames_patch
You can just backup and restore the saves in the Savegames_patch folder (or Savegames folder if you have old saves from more than a couple game updates ago, when the different folder was added), or the entire 'Larian Studios' folder.