Support and target are different things. Games are generally not targeting high-end system specification.
Yes, on PC the target varies depending on developer. For instance, Blizzard still targets Dx9 systems, but throws in extras for Dx11 users. On the flip side Avalanche targeted Dx11 systems on PC with their last game. Digital Extremes (Warframe) has moved their lowest supported system to Dx10 (Dx9 no longer supported) but their target is Dx11.
Most developers will target Dx9 or Dx11 and almost no one is going to target Dx12 but they may offer Dx11 (in case of Dx9 target) or Dx12 features (experimentation). Hence why many "Dx12" games are handicapped by a rendering path that is optimized for Dx11 rather than Dx12.
Digital Extreme's (as well as developers like CCP Games - Eve Online, Square Enix - FFXIV) are interesting to look at in terms of a changing platform. Since they know their games are going to be heavily used over the course of multiple hardware changes, they are constantly upgrading their engines. Eve Online started with Dx8 and now uses up to Dx11. Warframe started at Dx9 and now only supports Dx10/11. Square Enix started out limited to Dx9 but has moved to Dx11 on the PC (with some very nice results) which they are looking at attempting to port to Consoles.
Looking at Single Player games. CD Projekt Red (Witcher series) in the past have updated the engine for their Witcher games (The Witcher and The Witcher 2) by targeting a newer version of DirectX (enhanced editions) which required some non-negligible work for free. It remains to be seen if they do it for The Witcher 3, but don't be surprised if the Witcher 3 gets a significant engine upgrade when either Neo or more likely Scorpio (so both consoles can be supported) comes out.
CD Projekt Red is the poster of child of a developer that is able and willing to upgrade the engine for their single player game with hardware upgrades that come out before their next game is ready.
But it certainly isn't rare for games to get graphics upgrades due to the changing hardware landscape on PC. And I don't see a reason why that couldn't or wouldn't happen on consoles as well.
The Witcher and The Witcher 2 despite offering the engine upgrade as free to existing owners of the game still gained a significant number of new sales when their released the enhanced upgrades. Unfortunately, I don't think the majority of console developers or publishers will follow that model. Instead it's likely they'll go the Metro, Uncharted, Tomb Raider, Last of Us, GTA V, Darks Souls 2, etc. route of charging users again for an updated version even if they already own the game. But we can always hope that isn't the case and they'll go with how CD Projekt Red does it.
Regards,
SB