Do any of the DLC's add new things to the base game?

[ぐー]

Or are they separate stories?

[Alcator]

All paid DLCs add new things. Two of them ALSO are new stories.

The Rifts is ONE NEW Endless Mode map with ONE new buildable structure - a bridge. Initially, you are very space-constrained and there are no sites for unlimited resource collection - no drillable wall, no iron vein etc. - in that constrained space. You need to use available limited resources to construct bridge(s) to other island-like sections of the map where unlimited veins are located. To make up for it, the map features around 15 Steam Cores that can be collected as you expand.

The Last Autumn is the biggest expansion. It brings pre-freeze environment, and your objective is to construct the generator using brand new technologies, buildings and resources. As the scenario progresses, very different laws open up. Instead of Order vs. Faith, it's "Workers" vs. "Engineers", or "Productivity" vs. "Safety". Needless to say, once again, as you delve deeper in either branch, things get VERY UGLY. This DLC also adds new Endless mode - "Builders" - in which you try to build the generator and then survive as long as you can while the horrible winter with regular storms is already in progress. Surviving in -70°C without a generator is a tough challenge (although not as tough as Endurance mode).

On The Edge is a controversial scenario that puts you in charge of an outpost that does not have a generator, but can have Braziers (~ Steam Hubs, limited to only +2 heat zone max). The city that made you establish this outpost starts mistreating you, eventually leading to a sort of revolt. Without any source of food, you must establish trade routes with several other Settlements and trade resources for other resources. Eventually, a confrontation with the Main City has to be resolved. This DLC added a new mechanic to the Endless Mode - Settlements - which allows you to establish trade routes and trade with the Settlements. It also added quite a few new Random Hazards and new locations to the Frostlands.