The Legend Of Korra reveals what that world looks like years later. Republic City has its positives and negatives, but the world of The Legend Of Korra still has several influences from its predecessor.
Two of the most vital adults in the series, Tenzin and Lin, are the children of Aang, Katara, and Toph, respectively. The original character's journeys get references in flashbacks.
Azula
Azula did not become an essential character until season two of Avatar: The Last Airbender. Following Zuko's constant failures to capture the Avatar, the Fire Lord has his daughter search for Aang. Unlike her brother, Azula did not change her ways and see the opposing perspective. Instead, Azula found a single-minded determination in impressing her father and capturing the Avatar to guarantee a Fire Nation victory. Azula never makes an appearance during The Legend Of Korra, nor is she mentioned.
Sokka
As a member of the original trio, Sokka was essential from the beginning. While Katara can Water Bend, Sokka's strengths lie in strategizing. Over time, Sokka became better with a boomerang and a sword. In the first episode of The Legend Of Korra, Katara reveals that Sokka had already ed away. Sokka has one guest appearance during a flashback sequence, showing what became of him professionally.
Later on, Sokka is mentioned to have helped protect Korra as a baby. Otherwise, there is nothing personal revealed about Sokka. It is never shared if Sokka ever got married or had kids while the rest of his friends had.
Iroh
Iroh acted as a moral com of sorts for Zuko as they searched for the Avatar. Iroh continually tried to get Zuko to see other ways of life. In The Legend Of Korra, Iroh plays a similar role from the Spirit World. Although he had ed away, Iroh had left his body behind to live in the Spirit World. Korra interacts with Iroh numerous times when she visits, and Iroh has given her advice to help Korra get through her problem.
Zuko
Perhaps best known for his redemption arc, Zuko is a relevant part of The Legend Of Korra. After the credits rolled during the finale of Avatar: The Last Airbender, Zuko ed forces with Aang to create a better world than the one they had lived through. Between Zuko and Aang, they built Republic City, which has a statue of Zuko to honor him as a Fire Lord and his work in recreating the world. Zuko makes several appearances during The Legend of Korra. While he does not have massive participation in the fight sequences, his influence is shown in the new Fire Nation, lead by his daughter, Fire Lord Izumi, and grandson, General Iroh.
Katara
Katara appears multiple times during The Legend Of Korra. She took part in training Avatar Korra and was also known to use her abilities as a healer. Katara was an essential part of Korra's rehabilitation after the battle with Zaheer.
With Katara's help, Korra gained the strength to walk, and in other areas as well. Although Katara does not in the war efforts, her children do. Katara also spoke to Korra before the Avatar follows Tenzin to Republic City. Similar to Toph, Katara's impact was massive, but she did not participate in the fighting, leaving that to Korra and the new Team Avatar.
Toph
Toph does not appear as often as Katara in the present time, but between her appearance in season four and flashbacks of Toph, she is a very influential part of The Legend Of Korra. She invented Metal Bending as a child and became more useful as she helped Aang reshape the world. Toph's daughter, Lin, is the Chief of Police in Republic City, and several Earth Benders have learned how to Metal Bend. Toph also created the police, and her impact is always an essential part of the background of the world. Although she does not get involved in many fight sequences, Toph did guide Korra in the swamp and helped Lin, Bolin, and Opal rescue the rest of their family.
Aang
Korra's predecessor may not appear in physical form, but he does show up in the spiritual state. But, even without the connections to Aang's spirit, his impact is in every aspect of the series. There is a giant statue of Aang near Air Temple Island. Republic City only exists because of Aang and his work with Zuko. Aang had ensured that the next Avatar would receive help to master the elements like he never had. Besides, even if Aang is not there to help Korra, his son Tenzin, the lone Air Bending Master for years, has taken up that responsibility. Aang's Air Bending legacy continued in his youngest son, and later on, his grandchildren, Jinora, Ikki, and Meelo.