Were the successors of the prophets from their offspring?

Question:
Were the successors of the prophets from their offspring?
Answer:
Referring to the history of divine apostles, from Hazrat Adam (a.s.) to Hazrat Muhammad (s.a.w.) the last prophet, signifies that not only each of theprophets had a successor or successors for himself, but also those successors were from their offspring.
We briefly mention some instances here:
1- Shaith the successor of Hazrat Adam (a.s.)
A) Masudi says that: “When Hawwa became pregnant with Shaith, a light shone on her forehead, and when she gave birth to him that light was transferred to her son. And when he grew up completely, Hazrat Adam made him his successor and announced that he will be the argument of God after him and divine caliph on all the earth…” [1]
B) Ya'qubi says: “When Adam was going to die, he gathered his son, Shaith, and the sons of him and sent greeting to them and prayed for them… and willed his succession to Shaith and ordered him to be his successor among his sons. He ordered them to be pious and to worship Allah well and prohibited them from establishing communication with Qabil and his sons…” [2]
He also says: “Shaith began to perform his duties after the death of Adam (a.s.). He ordered his nation to be pious and to do good acts.” [3]
2- Sam the successor of Hazrat Noah
IbnAthir writes: “Noah willed his succession to his son Sam who was the eldest of his son.” [4]
Masudi writes: “God gave the leadership and heaven books to Sam, the son of Noah…” [5]
Ya'qubi writes: “Sam the son of Noah began to obey and worship God after his father…and at the time of his departure he willed his succession to his son Arfakhshad.” [6]
3- Ismail the successor of Ibrahim (a.s.)
Ya'qubi writes: “When Hazrat Ibrahim decided to leave Mecca, he willed staying beside the House of Allah (Ka'ba) and performing the rites of Hajj and teaching it to the people to his son Isma`il, and told him that: God will increase his children, make his offspring fruitful and grant blessing to his sons.” [7]
And Isma`il expressed his desire, at the time of his departure, to his brother Ishaq. By this way the will was transferred from father to son or to brother or to a member of family.
4- Sulaiman the successor of Dawud (a.s.)
HazratDawud expressed his desire to his son Sulaiman (a.s.) and told him that: “I am moving in the way of the whole people of the world … so act on the commands of your Lord and act on His agreements, promises and commands that have been mentioned in the Old Testament.” [8]
The Almighty God in this regard says:﴿وَلَقَدْ آتَيْنَا دَاوُودَ وَسُلَيْمَانَ عِلْمًا وَقَالَا الْحَمْدُ لِلَّهِ الَّذِي فَضَّلَنَا عَلَى كَثِيرٍ مِّنْ عِبَادِهِ الْمُؤْمِنِينَ * وَوَرِثَ سُلَيْمَانُ دَاوُودَ﴾“And certainly We gave knowledge to Dawood and Sulaiman, and they both said: praise is to Allah, Who has made us to excel many of His believing servants and Sulaiman was Dawood's heir” [9] [10]

Refrence:

[1] Muruj al-Zahab, vol. 1, p. 47 and 48, briefly
[2] TarikhYa'qubi, vol. 1, p. 7.
[3] TarikhYa'qubi, vol. 1, p. 8.
[4] KamilIbnAthir, vol. 1, p. 26.
[5] Akhbar al-Zaman, p. 75.
[6] TarikhYa'qubi, vol. 1, p. 17.
[7] TarikhYa'qubi, vol. 1, p. 28
[8] TarikhYa'qubi, vol. 1, p.57
[9] Naml, v. 15, 16.
[10] Ali AsgharRizwani, Imamology and answering to the doubts (1), p. 153.