itch.io

You didn't mention what platform you play on.  I am just another player and I'm no authority on this, but for what it's worth, this method has worked unerringly on every Ren'Py game I've ever played... on Windows/Linux: 

I download the whole game at every update. 

Unzip/extract the new one into it's own auto-named folder (matching the filename downloaded).

First, BEFORE I even run the new game, I go to the OLD version game folder, "game", and locate the "saves" folder in there, I right click that guy and select Copy.

Then I go to the NEW version game folder I just unzipped/unpacked into. I right click the folder named "game" and select Paste. 

In a few seconds, it copies every single saved game into the new install. Now I run the NEW version's .SH or .EXE and load up a couple of my favorite saves and immediately re-save over them as a means of confirming my favorites are saved in the newest format(just in case anything changed in the way it is written. 

After I play the new one for a bit and save again to a brand new slot and I'm convinced there is no issues, I exit the game and delete the OLD install completely.

It may seem lengthy, but it has been fool-proof for me so far.  Never a single lost saved game unless the Dev completely changed everything, but they are pretty good about warning of that beforehand.  And if you skip a version release when you update, it's not as guaranteed to work.

I hope that helps somebody out there.