The challenge: Have an expansionist empire conquer a region, resettle the people of that region into the center of their empire, and over time be assimilated by the conquered people in the capital region of the empire.
The only historical example I can think of this sort of bottom-up assimilation during the existence of its own empire is ancient Assyria.
In ancient Assyria, the conquest of Aramaean polities and the forced
resettlement of conquered peoples led to the accidental Aramaization of the formerly Akkadian-speaking empire. To the extent that Aramaic became the new dominant vernacular and the new lingua franca, while the language of official documents was an "Aramaic-Akkadian symbiosis". The Aramaization was thorough--the Babylonian and Persian Empires used Imperial Aramaic as their state languages, and almost all of the Mesopotamian region spoke Aramaic as vernacular centuries later and well into the early Islamic days.
In a broader sense, conquerors did sometimes assimilate to the majority culture of the people they conquered, but this didn't change the cultural map during the existence of the respective realms. While the Jin, Yuan, and Qing Dynasties adopted Chinese language and customs, Norman-descended nobles after centuries eventually adopted English, the Romans adopted Greek and Hellenism in the East, Turkic tribes adopted Persian, and so on... this only involved the conquerors moving to a new place and adopting its customs, rather than actually assimilating their own homeland accidentally with their own resettlement policy.
Creating the latter scenario is harder than it sounds. In most cases where an empire performed mass population transfers, there was some factor that prevented the relocated settlers, slaves, or serfs from leaving a lasting mark on the region, or the empire lasted long enough to undo the change during its rule. Rome had the coloniae and citizenship policy encouraging Romanization. European colonial empires and the trans-Atlantic slave trade captured from many areas and imposed a racial division in the colonies, while colonies were separated from metropole by the Atlantic. The Abbasid/al-Andalus slave trade and Saqaliba left little or no cultural mark because the slaves were castrated and were sold as eunuchs. The Inca Empire's mitmaq policy actually assisted the spread of Quechua because resettled populations were shuffled around and closely monitored. In all these cases, the empire(s) still controlled the dominant culture in its territory.
The closest situation that almost satisfies the challenge might be the Turkic slave-soldiers or ghulam in the Abbasid Caliphate, the descendants of whom eventually established the Turkic language in Azerbaijan where it evolved into Azerbaijani. However, this took centuries, and the process was not complete until the Safavid era after the Abbasids had fallen.