Kush & Luv were returned to Rama once Sita went underground. Actually, when Rama was furious and threatened to destroy the earth if Sita wasn't restored to him, Brahma convinced him to have the rest of the story (what was yet to happen) sung to the gathering, so Rama granted that. That night, he spent it in the same hut of leaves that Kush & Luv were living in during the yagna. After their rendition was over, they were handed over to Kaushalya, while the yagna continued. They served their family, and when Kaushalya, Sumitra and Kaikeyi died, they, their brothers and fathers were all there.
After the deaths of their grandmothers, Bharat was sent to conquer Gandhara with his sons Taksha & Pushkal. After this conquest, Taksha was made ruler of Takshashila (today's Rawalpindi) and Pushkal Pushkalavati (today's Jalalabad). Rama then asked him to identify a kingdom for Lakshman's sons where the people's needs were simple and where they could do their hunting and other pleasures without disturbing the rishis, and Bharat suggested Karupathadesha. Rama, Lakshman, and Urmila's sons Chandraketu and Angad went on an expedition and conquered this land. A city called Chandrakant was founded for Chandraketu, and a city of Angavad was founded for Angad.
After the return of Rama & Lakshman to Ayodhya, the visit of Kal-deva took place that ultimately resulted in Lakshman giving up his life, and Rama deciding to follow. He asked Bharat to succeed him, but there was no way Bharat was going to rule Ayodhya again. Instead, he told Rama that Kush and Luv had come of age, and that they should be crowned. They were, and then Shatrughan was sent a message. He gave his kingdom Mathura to Subahu, and another part of his kingdom Vidisha was given to Shatrughatee. After that, he returned to Ayodhya, and they and their kinsmen invoked the Pushpak and returned to Vaikuntha.
One interesting thing here - in the Raghukul dynasty, it was the tradition of only the eldest son to sit on the throne, as happened with Rama, but this was a tradition that Rama didn't seem to like. So not only did he put Shatrughan on a throne - he made sure that each of his sons & nephews - not just the eldest 4 - got a kingdom. Interesting precedent that was.