What happened to Luv Kush after Mata Sita's exile from erath? Please cast some light on thier cousins as well.

After several years from Mata Sita's exile, Bharat's uncle Yudhajit invited Rama to conquer Gandhara on his behalf.  Rama sent Bharat & his 2 sons Taksha & Pushkal to conquer the territory, while after that, he, Lakshman & Lakshman's 2 sons Angad & Chandraketu invaded Karupadhadesh (on the foothills of the Himalayas) and conquered that.  Bharat's & Lakshman's sons were given their own city kingdoms.

After that was the Durvasa incident which resulted in Lakshman's abandoning the world, and following that, Rama decided to abdicate the throne & follow him.  He offered the throne to Bharat again, but this time, Bharat would have none of it, and instead suggested that Kush & Luv be given North & South Kaushal Pardesh.  The two of them were crowned in Ayodhya, while in Mathura, Shatrughan too passed his kingdoms of Mathura & Vidisha to his 2 sons Subahu and Shatrughati.

Bharat's sons ruled Gandhara - their capitals were Takshashila (Taxila) and Pushkaravati (near Jalalabad). Lakshma's sons ruled Karupadhadesha - their capitals were Andavad & Chandrakant - likely in Nepal. Shatughan's sons Subahu and Shatrughati, had capitals in Mathura & Vidisha.