Unfortunately, I can't fully comprehend Paul's theology, and give you all the scriptures he hyperlinks to. But I am pretty sure that the aftermath of Absalom's rebellion is the clearest picture of what the apostle talks about here.
I already told you in my answer to your previous question, (Is there a difference between Israel of the flesh and the Israel of God?) that Ephraim represents the gentiles. Jacob prophesied in Genesis 48:19 that in him would be the fullness of the gentiles (מְלֹֽא־הַגּוֹיִֽם melo hagoim). You must have noticed that the history of the Twelve Tribes from the time of Jacob up to the Assyrian captivity was dominated by two of them: Judah and Ephraim/Joseph. The influence of the Ephraimites on the other nine tribes after Solomon's death was so great that the whole kingdom was sometimes called Ephraim. Jesus Himself represents both of them too. He already came as the suffering servant (Messiah ben Joseph), and is coming back as the king of the whole creation (Messiah ben David). We have to keep in mind that according to Jewish tradition (has been kept for the last two millennia) Ephraim and Manasseh wouldn't be considered Jews today, because their mother was an Egyptian.
Fast-forward to the Assyrian exile. The aristocracy and most people of any importance were forcibly relocated to other parts of the empire. They were mainly from the heart of the Northern Kingdom, the land of Ephraim and Manasseh. Unlike the Judeans, they lost their identity, mingled with the gentiles and vanished. From now on, we can't talk about the pure blood descendants of the ten lost tribes, because their DNA has been compromised. But since every believing gentile may have some in him/her, they can be considered descendants of Ephraim in the spiritual sense.
Back in Samaria, the left behind by the Assyrians mingled with foreigners that were resettled there from Mesopotamia. The descendants of those "bastardised" Israelites (at least in the eyes of the Jews), are known today as the Samaritans. They are quite important in the history of spreading the gospel, as they were the mid-stage (half blood) between the Jews (pure blood) and the gentiles (low percentage DNA).
In Deuteronomy 32 Moses predicted the unfaithfulness of the people and prophesied about the judgement and the exiles. He wrote:
18 You were unmindful of the Rock that boree you, and you forgot the
God who gave you birth. 19 “The Lord saw it and spurned them, because
of the provocation of his sons and his daughters. 20 And he said, ‘I
will hide my face from them; I will see what their end will be, for
they are a perverse generation, children in whom is no faithfulness.
21 They have made me jealous with what is no god; they have provoked
me to anger with their idols. So I will make them jealous with those
who are no people (בְּלֹא־עָ֔ם); I will provoke them to anger with a
foolish nation.
Wait a minute. I know that phrase from somewhere else. Yes, Hosea used it in the context of Ephraim (used in the book 37 times), the Ten Northern Tribes:
2 When the Lord first spoke through Hosea, the Lord said to Hosea,
“Go, take to yourself a wife of whoredom and have children of
whoredom, for the land commits great whoredom by forsaking the Lord.”
3 So he went and took Gomer, the daughter of Diblaim, and she
conceived and bore him a son. 4 And the Lord said to him, “Call his
name Jezreel, for in just a little while I will punish the house of
Jehu for the blood of Jezreel, and I will put an end to the kingdom of
the house of Israel. 5 And on that day I will break the bow of Israel
in the Valley of Jezreel.” 6 She conceived again and bore a daughter.
And the Lord said to him, “Call her name No Mercy, for I will no more
have mercy on the house of Israel, to forgive them at all. 7 But I
will have mercy on the house of Judah, and I will save them by the
Lord their God. I will not save them by bow or by sword or by war or
by horses or by horsemen.” 8 When she had weaned No Mercy, she
conceived and bore a son. 9 And the Lord said, “Call his name Not My
People (לֹא עַמִּי), for you are not my people, and I am not your
God.” 10 Yet the number of the children of Israel shall be like the
sand of the sea, which cannot be measured or numbered. And in the
place where it was said to them, “You are not my people,” it shall be
said to them, “Childrene of the living God.” 11 And the children of
Judah and the children of Israel shall be gathered together, and they
shall appoint for themselves one head. And they shall go up from the
land, for great shall be the day of Jezreel.
What is interesting, is the fact that the prophet talks about the Ephraim's return from the exile again:
10 They shall go after the Lord; he will roar like a lion; when he
roars, his children shall come trembling from the west; 11 they shall
come trembling like birds from Egypt, and like doves from the land of
Assyria, and I will return them to their homes, declares the Lord.
(Hosea 11)
So they were exiled to Assyria, but will return also from the west and the south? From everywhere? Very puzzling.
And what is even more interesting is the fact that both, Paul and Peter quote Hosea 1 in connection with the gentiles saying that this scripture has been fulfilled in them:
22 What if God, desiring to show his wrath and to make known his
power, has endured with much patience vessels of wrath prepared for
destruction, 23 in order to make known the riches of his glory for
vessels of mercy, which he has prepared beforehand for glory— 24 even
us whom he has called, not from the Jews only but also from the
Gentiles? 25 As indeed he says in Hosea, “Those who were not my people
I will call ‘my people,’ and her who was not beloved I will call
‘beloved.’” 26 “And in the very place where it was said to them, ‘You
are not my people,’ there they will be called ‘sons of the living
God.’” (Romans 9)
9 But you are a chosen race, a royal priesthood, a holy nation, a
people for his own possession, that you may proclaim the excellencies
of him who called you out of darkness into his marvelous light. 10
Once you were not a people, but now you are God’s people; once you had
not received mercy, but now you have received mercy. (1 Peter 2)
So now we only have two options before us. The two apostles were either very creative and twisted the scriptures to fit their agenda, or the gentiles really are the spiritual descendants of the divorced Ephraim of the flesh. John the apostle says the latter winking at us while writing:
Jesus therefore no longer walked openly among the Jews, but went from
there to the region near the wilderness, to a town called Ephraim, and
there he stayed with the disciples. (John 11:54)
Some see the two beams of the cross as the fulfilment of Ezekiel 37:15-28. It may be so, since the unification of Judah with the Ephraim is a big part of the messianic profile.
Now back to the rebellion of Absalom. The twelve tribes of Israel are united after the control of the kingdom returns to the father.
Handsome and vain Absalom (the father of peace, in reality more like the father of deception) as the usurper of the throne of David is a picture of the satan himself. In his death, however, he becomes the antitype of Jesus. He rode to his death on the mule, an infertile beast, trying to escape from the battlefield. The rebellious son hangs on a tree and is pierced three times, which brings peace to the whole nation. Jesus the obedient son wilfully rode to his death on an ass and her colt (more likely the symbol of the Jews and the gentiles again). The prince of peace brigs everlasting peace to the whole creation. The good news about the victory and restoration of the king (euaggelion in LXX: 2 Sam 18:19, 20, 26, 31) is brought by two messengers (the image reused by Isaiah in 40:9 & 52:7). An Israelite first and then a gentile from Kush. The delegation of all the tribes of Israel except Judah comes to Machanaim and ask David to be their king again. Mahanaim? I know that name. Wasn't this the place where Jacob split his family into two camps to save Joseph and Rachel? Coincidence? I don't think so.
9 And throughout all the tribes of Israel there was much discussion
and argument going on. The people were saying, “The king rescued us
from our enemies and saved us from the Philistines, but Absalom chased
him out of the country. 10 Now Absalom, whom we anointed to rule over
us, is dead. Why not ask David to come back and be our king again?” 11
Then King David sent Zadok and Abiathar, the priests, to say to the
elders of Judah, “Why are you the last ones to welcome back the king
into his palace? For I have heard that all Israel is ready. 12 You are
my relatives, my own tribe, my own flesh and blood! So why are you the
last ones to welcome back the king?” 13 And David told them to tell
Amasa, “Since you are my own flesh and blood, like Joab, may God
strike me and even kill me if I do not appoint you as commander of my
army in his place.” 14 Then Amasa convinced all the men of Judah, and
they responded unanimously. They sent word to the king, “Return to us,
and bring back all who are with you.” 15 So the king started back to
Jerusalem. And when he arrived at the Jordan River, the people of
Judah came to Gilgal to meet him and escort him across the river. 16
Shimei son of Gera, the man from Bahurim in Benjamin, hurried across
with the men of Judah to welcome King David. 17 A thousand other men
from the tribe of Benjamin were with him, including Ziba, the chief
servant of the house of Saul, and Ziba’s fifteen sons and twenty
servants. They rushed down to the Jordan to meet the king. 18 They
crossed the shallows of the Jordan to bring the king’s household
across the river, helping him in every way they could. (2 Samuel 19)
All in all, the whole Israel Paul talks about must be the gentiles (spiritual descendants of Ephraim, the Ten Lost Tribes) and the Jews.