A few examples:
4th Hokage: He was nominated and chosen when 3rd Hokage was still alive and was capable of fighting (courtesy: 9-tails attack on Konoha)
5th Hokage: after the Third Hokage deceased.
4th Raikage: After the Third Raikage deceased.
I thought the next Kage would only be chosen after the previous Kage's death. But that wasn't the case for 4th Hokage! Was he strong enough to be believed to take over Third? Was it because the Third believed he was too old ?
Btw, the Third continued to be Hokage after 4th was deceased!
I have never come across an info that notified the 4th Hokage's nomination..