The Ohr Hachaim Hakadosh finds a reference to the oaths in the first Tochacha: “And I will scatter you among the nations, and I will draw out a sword after you; and your land will be desolate, and your cities waste” (Vayikra 26:32-33). “I will scatter” refers to the first oath, which mandates that the Jews remain scattered and not come up as a wall, with a strong hand, to resettle Eretz Yisroel. “Among the nations” refers to the second oath, which mandates that the Jews remain submissive to the nations and not rebel against their authority. “I will draw out a sword after you” means that G-d will enforce the oaths: if the Jews violate them, they will die by the sword, G-d forbid, as the Gemara says (Kesubos 111a), “If you keep the oaths, good, but if not I will permit your flesh like the gazelles and deer of the field.” The final words “and your land will be desolate…” are the reason for the oaths: G-d wants the Holy Land empty in order that it should rest and make up for the Shmittah years that were not observed. We see here that the Ohr Hachaim says that the purpose of the oath is that the land be desolate and empty. The oath has nothing to do with the nations, and so the nations’ permission makes no difference.
The Ohr Hachaim comments that the book of Shemos, which describes the exile, begins with the word "and" alluding to Avraham, Yitzchok and Yaakov who were the first to live during exile. (The promised 400 years of exile began with the birth of Yitzchok.) Just as the Avos accepted the exile, these twelve sons accepted it as the decree of the King, unlike Esav, who moved to the land of Seir in order to avoid the decree (Bereishis 36:4 with Midrash Rabbah 84:2).
This also provides an answer to Rashi's question: Why does the Torah repeat the names of the twelve sons of Yaakov who came down to Egypt? It already listed them in Bereishis 46. According to the Ohr Hachaim, the answer is that the Torah is making a point of listing those who willingly accepted the exile.
The rest of the words of the verse fit in with this theme: "Who were coming" is in the present tense to indicate that even if they had not been forced to come, they would have come willingly. "With Yaakov" indicates that they were similar to Yaakov in their willingness to accept the exile. "Each man with his household they came" is the proof that they were accepting exile, for if they had come to Egypt for some temporary personal reason, they would not have uprooted their entire households from Canaan.
The Ohr Hachaim Hakadosh in his commentary Rishon Letzion explains the meaning of the verse in Eicha 1:7, “When her people fell into the hands of the enemy, and she had no helper, the enemies saw her and laughed at her end.” When the Babylonians besieged the city and Yirmiyahu announced clearly that it would fall into their hands, and similarly when the Romans besieged the city and Rabbi Yochanan ben Zakai announced that it would fall into their hands, the Jewish people had no helper, i.e. they did not take the advice of their helpers, their sages. Hashem gave them a chance to save themselves and the Temple by surrendering to the enemies, but instead they fought back and lost everything. “The enemies laughed at her end” – at the fact that everything came to an end and the Jews lost their city and the Temple.
The next verse continues on the same theme: “Jerusalem sinned a sin; therefore she became an outcast.” Hashem’s decree was only that the Jews should be subservient to Babylon, not necessarily that they should go into exile. Hashem would have found a way to punish them in their land. But because Jerusalem stubbornly refused to give in, the destruction of the Temple resulted, says the Ohr Hachaim.
In his commentary Rishon Letzion on Shir Hashirim, the Ohr Hachaim Hakadosh connects the oath in Shir Hashirim 2:7 with the previous verse, 2:6: “His left hand is under my head, and His right hand embraces me.” The “left hand” of Hashem is an expression for His attribute of strict justice and retribution, while His “right hand” signifies mercy. We find the same symbolism used by Rashi on Shemos 15:6, who says that when we do Hashem's will, even His left hand becomes a right hand. In Sotah 47a we find that a person as well “pushes away [a child or disciple] with his left hand and brings near with his right hand.” Here too, in exile Hashem chastised the Jewish people, but with His right hand he embraces them by allowing them to attain great heights of Torah knowledge.
Perhaps the Jews will feel abandoned in exile and say, “Although we are studying Torah, we are suffering and it seems as though Hashem is ignoring us.” Therefore Scripture says, “I adjure you, daughters of Jerusalem…” Do not force the hour and do not arouse the love, for perhaps Hashem does not want to bring the redemption now. When Hashem desires the love, He will rush to bring it.
This explains the next verse, 2:8: “The voice of my Beloved, behold it has come…” It is so certain that Hashem will fulfill His promises to us that it is considered as if it already happened.