MBJ. Sources

Other Peoples Work on the Messiah son of Joseph

EXTRACTS

https://kingdavidkabbalah.com/mashiach-ben-yosef-mashiach-ben-dovid/
In connection, while in the desert Am Yisrael was divided into four groups. The tribe of Yehuda led two tribes under his tribe flag which was a picture of a lion. There were another two camps and then the last camp which is the flag tribe Efraim; descendants of Yosef, his flag is a symbol of an ox. In the blessing from Yakov to Yosef, he was chosen to be the firstborn and is symbolized by an ox, the sign of strength with horns to gore the opposing nations. Here we see that in the front leading is the tribe of Yehudah and then Yosef to gather the nation. The Levi-im; representing the Shechinah, in the center. This structure of the tribes of Israel in the desert is the spiritual structure of the Am Yisrael.

https://www.chabad.org/library/article_cdo/aid/101747/jewish/Mashiach-ben-Yossef.htm
The essential task of Mashiach ben Yossef is to act as precursor to Mashiach ben David: he will prepare the world for the coming of the final redeemer. Different sources attribute to him different functions, some even charging him with tasks traditionally associated with Mashiach ben David (such as the ingathering of the exiles, the rebuilding of the Bet Hamikdash, and so forth).4 

The principal and final function ascribed to Mashiach ben Yossef is of political and military nature. He shall wage war against the forces of evil that oppress Israel. More specifically, he will do battle against Edom, the descendants of Esau.5 Edom is the comprehensive designation of the enemies of Israel,6 and it will be crushed through the progeny of Joseph. Thus it was prophesied of old, "The House of Jacob will be a fire and the House of Joseph a flame, and the House of Esau for stubble.." (Obadiah 1:18): "the progeny of Esau shall be delivered only into the hands of the progeny of Joseph."7

This ultimate confrontation between Joseph and Esau is alluded already in the very birth of Joseph when his mother Rachel exclaimed, "G-d has taken away my disgrace" (Genesis 30:23): with prophetic vision she foresaw that an "anointed savior" will descend from Joseph and that he will remove the disgrace of Israel.8 In this context she called his name "Yossef, saying 'yossef Hashem - may G-d add to me ben acher (lit., another son), i.e., ben acharono shel olam - one who will be at the end of the world's time,'9 from which it follows that 'meshu'ach milchamah - one anointed for battle' will descend from Joseph."10

The immediate results of this war11 will be disastrous: Mashiach ben Yossef will be killed. This is described in the prophecy of Zechariah, who says of this tragedy that "they shall mourn him as one mourns for an only child." (Zechariah 12:10).12 His death will be followed by a period of great calamities. These new tribulations shall be the final test for Israel, and shortly thereafter Mashiach ben David shall come, avenge his death, resurrect him, and inaugurate the Messianic era of everlasting peace and bliss.13

This, in brief, is the general perception of the "second Mashiach," the descendant of Joseph through the tribe of Ephraim.

Quite significantly, R. Saadiah Gaon (one of the few to elaborate on the role of Mashiach ben Yossef) notes that this sequence is not definite but contingent! Mashiach ben Yossef will not have to appear before Mashiach ben David, nor will the activities attributed to him or his death have to occur. All depends on the spiritual condition of the Jewish people at the time the redemption is to take place:

This may explain why Rambam does not mention anything about Mashiach ben Yossef. R. Saadiah Gaon18 and R. Hai Gaon,19 as well as a good number of commentators, do refer to him briefly or at length. In view of the divergent Midrashim and interpretations on this subject it is practically impossible to present a more definitive synopsis that would go far beyond the above. Thus it is wisest to cite and follow R. Chasdai Crescas who states that "no certain knowledge can be derived from the interpretations of the prophecies about Mashiach ben Yossef, nor from the statements about him by some of the Geonim;" there is no point, therefore, in elaborating on the subject.20

https://en.wikipedia.org/wiki/Messiah_ben_Joseph
Throughout the Hebrew bible Ephraim is often used to refer collectively to the northern kingdom. Ephraim was the leading tribe in the north. It has been claimed that Messiah ben Joseph does not represent the leader of the Ten Lost Tribes and that he is never presented as such.[43] Rather he is presented as the leader of all of Israel. However some later Jewish sources do explicitly call the Messiah ben Joseph the leader of the Ten Lost Tribes.

Mikweh Israel ["Hope of Israel"] was written by the 17th-century kabbalist Menasseh Ben Israel. The text deals at some length with the author's theory that parts of the ten tribes can be found among the Native Americans.[45]: 17-56 In the text the author calls the Messiah ben Joseph the future leader of the ten lost tribes.[45]: 43

In his commentary on Ezekiel 37 the Malbim also says that the Messiah ben Joseph will be the leader of the Ten Lost Tribes when they return. The Messiah ben Joseph will initiate union with Judah, who will be led by the Messiah ben David. Later, the Messiah ben Joseph is killed and Messiah ben David will rule over all Twelve Tribes..
Some academic scholars[who?] have argued that the idea of two messiahs, one suffering, the second fulfilling the traditional messianic role, was normative to ancient Judaism, in fact predating Jesus. Early Christians (who were Jews) might have viewed Jesus as fulfilling this role.

See Also:
Who is the Moshiach ben Yosef?
by Moseh Shulman,
https://jewsforjudaism.org/knowledge/articles/moshiach-ben-yosef/