In Mosiah 2:29–30, King Benjamin stated, "I say unto you that I have caused that ye should assemble yourselves together, that I might declare unto you that I can no longer be your teacher, nor your king; For even at this time … I should declare unto you this day, that my son Mosiah is a king and a ruler over you."
That is an official proclamation announcing the new king. At this point, there may have been some kind of investiture, where Mosiah would come up and perhaps be given the brass plates, or the Sword of Laban. After some interruption, King Benjamin resumes in verse 31 with the words, "And now, my brethren."
In most successions from one king to another, what happens? The king dies and his sons fight over who is going to be the next king. It is very unusual for a king to appoint a son while the king is still alive. Now, there will be a co-regency for the next three years, with Benjamin and his son working together. At that point, King Benjamin will die, but in the meantime, he has provided for a smooth transition from himself to Mosiah, and Mosiah will successfully reign for thirty-three years without war or disruption, as far as we know.
Stephen D. Ricks, "Kingship, Coronation, and Covenant in Mosiah 1–6," in King Benjamin’s Speech: "That Ye May Learn Wisdom," ed. John W. Welch and Stephen D. Ricks (Provo, UT: FARMS, 1998), 233–276.