itch.io

Assuming you're on PC: In the file folder containing the old version of the game, go to the "game" folder and locate the "saves" folder. Then, go to the "game" folder of your NEW downloaded version (which shouldn't have a saves folder inside) and create a new folder, naming it "saves" exactly the same as the other one. When that's done, just copy/paste the entire contents of the old saves folder into the new one and you should be set.

Assuming you're not on PC: I got nothing.

You don't need to manually move your save files over when you update - the game will do it for you :)  Just make sure to NEVER delete the app if you're on Android or all your saves will disappear as well! Just install the new apk over the old one.