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That depends on the game engine.
Many games are made with an engine called Ren'py.
For those games, you can find the savegames in a folder called "c:\XXXgamenameABC\game\saves", where c:\ is the drive letter and XXXgamenameABC the folder for that game.
Ren'py also saves all savegames in the user directory in Windows.
So if you download a new version, you can manually copy your savegames to the new folder, or depend on the savegames in your user directory, unless you regularly clean it (as I do).
After copying the saves, you can delete the older version. Although in some games, the developer makes some drastic choices in the story line, deleting some older content and leaving you to prefer the old version. In that case, you could choose to keep both.
Other games are made with different engines, in that case, the save games may be in a different location.
You can recognise the engine by looking at the colored label to the left of the game name in the forum:
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This is an example of a Ren'py game:
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