[Jeremiah 3:18] IN THOSE DAYS THE HOUSE OF JUDAH SHALL WALK WITH THE HOUSE OF ISRAEL, AND THEY SHALL COME TOGETHER OUT OF THE LAND OF THE NORTH TO THE LAND THAT I HAVE GIVEN FOR AN INHERITANCE UNTO YOUR FATHERS.
The Midrash says:
“Behold thou art fair, my love, yea pleasant, also our bed is green” (Song of Solomon 1;16).
Midrash: ‘“Israel says: Behold thou art fair, my love When you will take retribution from the worshippers of idols. Yea pleasant when you pay the reward of those who fear you. Also our bed…: These are the “Ten Tribes.” [In Hebrew bed is “eres” while ten is “eser” and uses the same letters, and “our bed” (erseynu) sounds like “our ten” (esereynu).] They are those who were exiled beyond the Sambation River. The exiles of Judah and Benyamin [i.e. the present-day “Jews”] are destined to go unto them and bring them back in order to merit with them the Messianic Era and life in the World-To-Come. This is as it says, “In those days the house of Judah shall go unto the house of Israel and they shall come together out of the land of the north unto the land that I have given for an inheritance unto your fathers” (Jeremiah 3;18). ‘
-Yalkut Shimeoni, Song of Solomon 905.
The verse should be read according to the above in accordance with the Hebrew as:
“THE HOUSE OF JUDAH SHALL GO UNTO [rather than “WALK WITH”] THE HOUSE OF ISRAEL”.
People from Judah and Benjamin (i.e. legal Jews) shall go unto the Lost Ten Tribes who are beyond the Sambation River. In our works (“Ephraim” and “The Tribes”) we explained that the Sambation River means Israelite Exiles who migrated to the west. The simple meaning is that Jews from Judah will go unto the Lost Tribes to bring them back. It could be said that the Jewish members of Brit-Am Israel is doing this to some degree.
Concerning Jeremiah 31:8 see what we wrote in our commentary to Jeremiah:
[Ezekiel 37:21] AND SAY UNTO THEM, THUS SAITH THE LORD GOD; BEHOLD, I WILL TAKE THE CHILDREN OF ISRAEL FROM AMONG THE HEATHEN, WHITHER THEY BE GONE, AND WILL GATHER THEM ON EVERY SIDE, AND BRING THEM INTO THEIR OWN LAND:
[Ezekiel 37:22] AND I WILL MAKE THEM ONE NATION IN THE LAND UPON THE MOUNTAINS OF ISRAEL; AND ONE KING SHALL BE KING TO THEM ALL: AND THEY SHALL BE NO MORE TWO NATIONS, NEITHER SHALL THEY BE DIVIDED INTO TWO KINGDOMS ANY MORE AT ALL.
“NEITHER SHALL THEY BE DIVIDED INTO TWO KINGDOMS ANY MORE AT ALL”: but until the end times they will have been so divided.
Here the Prophet is referring to the return of BOTH Judah and “Israel” and this includes descendants of both who are scattered in other countries AS WELL as those concentrated in specific nations.