If you use Steam, Epic/Heroic, Rockstar, etc. launchers, then cloud-syncing should already be inbuilt. However, locally installed games, such as those downloaded from itch.io, don't have this functionality out-of-the-box. Instead, we can try to implement cloud saving through Git.

Clone this repository into where your game save is located.

git clone https://github.com/Wartybix/git-cloud-sync

Change the remote origin URL to point to your own forked repository:

git remote set-url origin https://github.com/your-username/your-game-cloud-save

Alternatively you can fork this repository in GitHub first, and just clone your forked repository, in which case you shouldn't need to change the origin URL afterwards.

Change the upload-script.sh script to include the save file(s) of your game where the commented line is.

In your game's advanced options in Lutris, enter the path of download-script.sh into the Pre-launch script field. Set Wait for pre-launch script completion to on. Enter the path of upload-script.sh into the Post-exit script field.

Incoming changes should be pulled from the remote repository just before the game is opened, and local changes should be pushed immediately after closing the game.

If you are not using Lutris to open your locally-installed games, you can instead change your game's .desktop file to point to game-wrapper.sh for its exec field. Open game-wrapper.sh and add the path of the game's executable where the commented line is.