Lots of recent and upcoming games support upgrades from the PS4 to the PS5 versions, but your progress won't necessarily come with you.
The next generation is here, and thanks to backwards compatibility and upgrades, it will also be a soft continuation of the current one. In the case of the PlayStation 5, though, it isn't quite as simple as you might think. You may expect to just boot up your new PS5 and have it retain any progress from your old PS4 games, but that won't necessarily work.
The confusion is compounded by the fact that playing PS4 games on your PS5 comes in two distinct methods. The first is simply playing backwards compatible games on your PS5, while the second is upgrading your PS4 version to a PS5 version. So will your old saves work if you're playing a PS4 backwards compatible game on PS5? Will it work if you upgrade? The answer to one doesn't always apply to the other.
There have also been some reports of version confusion, with players downloading the PS4 version onto their PS5 by mistake. Sony recently issued an update to mitigate this by adding alerts, but it's still not the most simplified solution. On Xbox Series X/S, by comparison, Smart Delivery essentially treats games as the same across platforms. We've seen at least two specific instances already in which developers have said the Xbox saves will transfer, but PlayStation ones won't.
The shared progression should apply to saves as well, though we'll have to see both consoles in the wild to really tell how they compare in ease-of-use and importing saves. It's also very possible that developers will create new solutions for the cross-save issue sometime after launch.
Here's what some studios have already said about PS4-to-PS5 cross-saves--and some answers are clearer and more satisfying than others.
Call of Duty: Black Ops Cold War
Call of Duty: Black Ops Cold War offers shared progression across generations, including all of your gear and battle pass unlocks. But unlike some others, it's not free. You'll either need to buy an upgrade or the cross-gen bundle to play the PS5 version.
Codemasters' Dirt 5 will not allow a transfer of your saved progression from PS4 to PS5. This was revealed in a response to a fan on Twitter, which said you will be able to move across any of your Playground track creations, but you can't carry any in-game progress, currency, or custom liveries. However, these are all supported when transferring from Xbox One to Series S or Series X.
Currently on Xbox, all progress can be carried over between generations. On PlayStation, your Playgrounds creations can be carried over, but other game progress (Career, currency, saved liveries) cannot. If that changes, we'll let you know!— ONE MONTH TO GO #DIRT5 (@dirtgame) September 24, 2020
Tripwire Interactive announced its shark game Maneater would get enhanced with a free next-gen upgrade, including 4K at 60 FPS, DualSense support, and more. But if you do take advantage of the offer, you'll have to start anew. In a FAQ for the enhanced version, the studio notes, "This is a separate application/game on PlayStation 5 and Trophies, stats and saves will not transfer." The same FAQ notes that on Xbox Series S and Series X, it is not a different application and so saves and Achievements will transfer.
Marvel's Avengers is one exception to many of the others, with a free PS5 upgrade that will also let you carry over your progress. Given that it's a live game--and that many players may have gotten the PS4 version for its exclusive access to Spider-Man--this is a must-have feature.
The PS5 got two Spider-Man games side-by-side--a remaster of the 2018 Spider-Man game, and the new spin-off Spider-Man: Miles Morales. This has led to some confusion, because the cross-save situation differs between the two. Spider-Man Remastered didn't allow you to import your saves at launch, but added post-launch patch added the funtionality. The update added an exporting feature to take your saves with you to the remastered version, but the exact method remained convoluted. You need to go into the PS4 version of the game to export your save, then separately import it into the PS5 version. This is technically possible on a single console by simply downloading both versions of the game, but it's not a simple process.
Spider-Man: Miles Morales
Spider-Man: Miles Morales, on the other hand, let you transfer your save to the PS5 version from the start, without the need for a post-launch patch. It also includes a free upgrade to the game itself, so if you simply purchase a PS5 sometime later you can still play Miles Morales with all the new hardware's bells and whistles.
Yakuza: Like a Dragon
Yakuza: Like a Dragon will get a PS4-to-PS5 upgrade, either by inserting the disc or owning the game digitally. However, PS5 players will have to wait--while the Xbox Series X/S version will launch on November 10 alongside the new consoles, PlayStation won't get its next-gen version until March 21. And while the Xbox version will transfer your data from the old console to the new one, the PlayStation versions won't.
To clarify, physical owners of Yakuza: Like a Dragon on PS4 can also use the original disc to upgrade to the PS5 version when it releases on March 2, 2021.— RGG Studio (@RGGStudio) September 28, 2020
Save data will not be transferable between the PS4 and PS5 versions of the game.