Hello! The easiest way would be to claim your free Steam key and play the game through Steam, which features automatic cloud saves.
The only other way is to copy the save file between the two platforms. It is located in these paths:
- Windows: %APPDATA%\crossing-chains\progress.dat
- Linux: ~/.local/share/crossing-chains/progress.dat
- Mac: ~/Library/Application Support/crossing-chains/progress.dat