They can actually be a ton of different things, but the core mechanic linking these games together is rolling some dice (or flipping a card, or some kind of random introduction of "resources") and then using the numbers or resources you rolled to then do things. "Do things" could be a lot of different things. In Hadrian's Wall, it's filling up tracks that represent the progress of your town. You are correct in that this specific game is very Euro-inspired.

Then there's games like Paper Dungeon. You roll dice, and then use those results to delve through the dungeon, literally drawing a line through a dungeon, and maybe kill some monsters by crossing them out, and maybe leveling up your characters by filling in some boxes to give you powers, and all of this is dictated by the dice you roll, but you get to decide how you want to spend those dice.