It's not like rivalry. Both Dhoni and Kohli had different preferences for the national team selection. Dhoni would prefer players such as Kedar Jadhav and Dinesh Karthik, who play for CSK or India Cements. On the other hand, Kohli is said to have a preference for RCB/Delhi players, such as Rishabh Pant and Yuzvendra Chahal. Also, both of them support opposing camps in the BCCI. Dhoni is considered as a N Srinivasan loyalist, while Kohli supports the other side.
For now more focus will be on Rishabh Pant as he is just 23 and Dinesh Karthik is already 35 so may play for another couple of years. But in last few years Dinesh Karthik has being a good finisher and have ability to maintain calm under pressure and can hit six on last balls of innings while chasing the score which not everybody can do it. Rishabh has a long way to go and if he can prove consistency and gather behind the wickets cleanly, he will have more often his place fixed in the team squad.