What follows is a detailed review of the eleventh chapter of the Gra's sefer, Even Shlema. The Gra himself is in bold, the notes on the sefer itself, by Rabbi Shmuel Maltzin are in bold italics. My own thoughts follow in regular print.
Chapter 11 - In the concepts of the Erev Rav and the birthpangs of Moshiach, and the concepts of the future redemption, may it be speedily in our days.
1. The four concepts of the redemption will be in the four points (**) on the calendar that the world is judged (Rosh Hashana 16). This is reference to Pesach, when there will be a redemption from the subjugation [of the nations of the world]; The final judgment of the enemies of Hashem will be on Rosh Hashana; On Shavuos (Atzeres), Moshe Rabbenu A"H (*) will accomplish the ingathering of the exiles; On Succos, the Beis Hamikdash will be rebuilt.
(*) It seems that he means the sons of Moshe (***), like we find in Raya Mehemna, section two (120:1), what did the sons of Moshe place... And in his explanation of Raya Mehemna 23:1, 4, 5 and ד' ענפין, as well as Shir Hashirim 6:5, where he says that they are referred to in the verse, "And saviors will come up on Mount Tzion..." And here in the Raya Mehemna the indication is that the redemption in Nissan will be through Moshiach ben Dovid, who corresponds to the attribute of chessed (kindness). The destruction of Esav will be through Moshiach ben Yosef, who corresponds to the attribute of din (strict justice), as we saw, that this is how the cleansing occurs... to burn the straw with the left side. To this is brought a proof from the verse, "The house of Yosef will be a flame and the house of Esav will be straw." This needs further insight, because in his explanation of the 'Savei d'bei Asuna,' in the piece about the מישרא דסכיני, he writes that although Moshiach ben Yosef will rule over the entire world and destroy all the evil ones, as the verse says, "With his horns he shall strike the nations..." However, Esav will not fall to him, but rather to Moshiach ben Dovid. It could be that that which he writes in Raya Mehemna, that the straw is burnt by the left side, is not referring to Moshiach ben Yosef himself. Rather, his intent is that Moshiach ben Dovid will make use of the attribute of Gevurah (restraint) of Moshiach ben Yosef, and through this, the evil will be destroyed. This is like it says, "And afterwards all is taken from Gevurah," and like it says in the verse, "The house of Yakov(****) will be like fire, and the house of Yosef will be like a flame..." See Raya Mehemna Pinchas 252:1; There the calf will pasture; one will remove the great Roman, the other will remove the lesser Roman. See there in his explanation 53:1, under the heading "מרא דעגלא."
(**) If we look at the mishna in Rosh Hashana, we can see the parallels between the judgments that occur each year and the way the day itself will be expressed in the Geulah process. Pesach is the first day mentioned in the mishna, and it is the day we are judged on the produce. As we will see later on in the Gra, the fact that we stop eating chametz in the sixth hour is to balance out the Erev Rav's power in the sixth hour. We also find that the Gemara talks about what prevents the Jewish people from fulfilling their mission, and it refers to the subjugation of the nations of the world as well as the "שאור שבעיסה" - a reference to the Evil Inclination. It would seem that the Erev Rav is also included in the category of things that prevent the Jewish people from forging their bond with Hashem. Thus these concepts are parallel, and it is the holiday of Pesach that is the Jewish people's time to balance out the power of these things which represent the forces of evil. The judgment on the wheat also clearly has to do with the sin of Adam, which caused him to have to produce bread by the sweat of his face. The Jewish people have difficulty completing their rectification as long as these other forces stand in their way. Pesach, which is the main time for the tikkun, will thus be the time that those forces are finally removed.
The second date of judgment mentioned in the mishna is the day of Shavuos. This is the time when we are judged on the fruits of the tree. The parallel here is more difficult to see, but perhaps the following could be said. The fruits of the tree represent the concept of Malchus, as we see in the arba minim, that the concept of malchus is found in the Esrog. This is because the fruit is like a womb, containing the seed. This corresponds to the concept of the ingathering of the exiles, which the Gra will soon say is completed in the second stage with the return of the tribe of Yehuda. This is representative of the stage of Malchus (Moshiach ben Dovid is from Yehuda). It is interesting that the Gra says that the ingathering will be accomplished by Moshe Rabbenu, and requires deeper analysis as to why he is the one who is charged with this task.
The third date in the mishna is the date of Rosh Hashana, when all the people of the world 'pass before Him like b'nei Maron.' This is the day of judgment every year, and is the day of final judgment for the nations in the redemption process, as well.
The fourth day is Succos, when the judgment occurs for the water - i.e. how much rain will fall that year. Rain is the medium through which heaven and earth are joined, and it is the means through which Hashem gives physical sustenance to the world. The Beis Hamikdash represents this very concept, and therefore it is built on Succos. Furthermore, Succos itself is the time of the connection between the male and female concepts, as is represented by the mitzvos of Succos - the Lulav and the Succah. The Succah is representative of the Shechina that the Jews experienced in the wilderness, and this is the very idea of the Beis Hamikdash.
(***) It is interesting that R' Maltzin understands that it is speaking about the sons of Moshe. From what I have heard from numerous sources, Moshe Rabbenu himself actually returns at the advent of Moshiach.
(****) Here it seems that R' Maltzin understands that Yakov is reference to Moshiach ben Dovid. This entire piece is difficult to understand, because we see explicitly in numerous places that Yosef is the counter-force of Esav, and that Esav only falls to the children of Rochel. Generally, Yakov and Yosef line up because Yakov is Tiferes and Yosef is Yesod. Yakov takes the blessings of the first born and gives them to Yosef in place of Esav. We also find a clear reference in Rashi when Yakov returns from Lavan's house. Another possible explanation of the apparent issue with the Gr'a in the Tana D'bei Asuna could be that Moshiach ben Yosef is the one who fights the war of Gog Umagog, fighting against Esav. He will not be able to actually complete the destruction of Esav on his own, but rather, will join forces with Moshiach ben Dovid, who will help him complete the task. This works out better in the passuk in Ovadiah, "And the house of Yakov will be fire, the house of Yosef a flame, and the house of Esav will be straw; they will light them and consume them. There will be nothing remaining of Esav, for Hashem has spoken." Here we see that Esav is destroyed by two parties, the house of Yakov and the house of Yosef, and they both take part in the destruction. If we assume that Yakov is reference to Moshiach ben Dovid and Yosef is reference to Moshiach ben Yosef, it is clear that both play an integral role in the final destruction of Esav.