Do save games carry over from PS4 to PS5 / Xbox One to Xbox Series X, etc.? 

Yes, save games do carry over from previous console generations.  

Xbox

When you start the game on Series X|S, the console will automatically detect a previous generation save game and offer to load it, allowing players to continue their game from where they left off on a previous generation console. For more information, refer to the Xbox support article Moving from Xbox One to Xbox Series X|S (Opens in a new tab)

Playstation

If you have a PlayStation 4 save game on your PlayStation 5, the game will auto-detect this and, upon booting up, ask if you want to continue from this save game.

To bring a save game from the PS4 to PS5, follow the steps and information in the PlayStation support article about How to transfer data from PS4 to PS5 (Opens in a new tab)

You cannot “downgrade” a save game, meaning you cannot use your PS5 save game to continue playing on the PS4.