How Ramayan transitioned into Mahabharat?

The kingdoms which were given to the post-Ram generation kids, they are all mostly the Mahajanapadas, which we see rise to prominence in the Budhha-Jain era.

Can anyone please help me on how this transitioned into the MB era? 😊

It didn't, exactly.  While Rama may have set up 8 kingdoms, 4 of them were clearly wiped out by the chandravansh - the Druhyus, - Mathura and Vidisha, belonging to Shatrughan's sons, and in Gandhara, Bharat's successors too were dethroned.  It's not known whether that happened immediately during Taksha, Pushkal, Subahu and Shatrughatee, or after they were gone.  Also, nothing was ever heard of again of the kingdoms belonging to Luv, Angad or Chandraketu.

Kush continued the Raghu dynasty, and it seems to have ended w/ Brihadbala.  Brihadbala was defeated twice - once during the original Rajasuya yagna by Bhima, and later during Karna's Vishmi conquests by Karna.  So in the Kurukshetra war, he joined the Kauravas and was often locked in battle w/ Abhimanyu.  On day 13, he was killed in battle by Abhimanyu shortly before the latter was disarmed and ultimately slain.

I think the Raghu dynasty ended w/ Brihadbala's death

So to clear it: post-Ramayan, the mahajanapadas slowly lost their importance, and that was again regained post-MB right?

You know, just now I was reading about Brihadbala! 😆