Do the comics in these box sets continue the narrative directly after the animated TV series finale?

Yes, absolutely. The comic trilogies collected in the Avatar: The Last Airbender box sets, especially the earliest ones like 'The Promise,' serve as direct narrative continuations of the animated series. They pick up immediately after the events of 'Avatar Aang' and explore the immediate aftermath of the Hundred Year War, the establishment of the United Republic, Zuko's personal quest to find his mother, and other significant plotlines and character developments.