"Time is convoluted in Lordran" was used for interesting things, that DS2 and BB barely touched on. It is not just a handwave to let your character do whatever.
For example, DS1 had some instances where you'd help a NPC, or be helped by one, only for you to find their corpse and equipment in a later level, which led to a ton of story reconstrucion when analyzing that lore. The use of this phrase is that, through the centuries, several undeads have tried to reignite the flame and replace gwyn, and you meet them through time convolution. The main and favorite example is Solaire, but it is far from the only one.
DS2 also has NPCs that advance their own story alongside yours. But I don't think it does the stuff where a character would be dead in one area that had helped you in another, with one big exception I think, that was interesting because you find their corpse before meeting them live. There's one or two NPCs that are trying to complete the game, like you, but none of them fails midway like in DS1. I do not think that it is mentioned that "time is convoluted in drangleic", so I am just assuming that it inherited that property from Lordran. Maybe things are different, but since the multiplayer looks the same, I assume it is based on the same idea.
Both Dark games let you try the "invasion" mechanic on NPCs, though, which in the Dark games is based on time convolution, while BB and Demon don't, although Demon does have a boss that can be an invader.
BB just plain old only has hunters be there for the multiplayer; the other NPC hunters you find are not there trying to do the hunt alongside you, they are from other factions or just have no explanation at all. The NPC summons are characters that have different motivations than yours.
Demon, I think, didn't do this, and the NPCs explicitly advance their plot alongside you; they are not necessarily trying to awaken the old one. But since it precedes Dark 1, I cannot fault it for it. Demon does something interesting, that may have been explained as "time is convoluted", in that in Black Tendency levels, the NPCs could appear as evil versions, but there's not really a lore explanation for this. It was still a cool touch. None of the later games has done anything interesting with the White/Black tendency stuff, barely trying to imitate it with the Ascetic system in DS2 and Insight in BB, which is far from the same.