Hindus believe in a cyclic nature of time. Here the same key events are repeated forever. Then we should have the same 'Chiranjeevi's born repeatedly. This means, we should have an infinite number of them now. Else, they should die at some point of time. When and how does this death happen?
Chiranjeevis live for one Kalpa, as described in this chapter of the Vishnu Purana:
By immortality is meant existence to the end of the Kalpa: life as long as the three regions (earth, sky, and heaven) last is called exemption from (reiterated) death. The consequences of acts of iniquity or piety, such as Brahmanicide or an Aśwamedha, endure for a similar period, or until the end of a Kalpa, when all within the interval between Dhruva and the earth is destroyed.
But there are some exceptions to that. For instance, Agastya has been blessed with a significantly longer life as I discuss here. Also, more generally the inhabitants of Maharloka survive the end of the Kalpa; see my answer here. But at least the standard kind of Chiranjeevi, who lives within the three worlds, perishes along with the three worlds at the end of the Kalpa.