This is mostly brought on by reading the excellent discussion stemming from /u/bitparity's mention of "Empire" as a loaded term in this week's Monday Methods post. That discussion also informs the use of 'polities' above.
I'm specifically not thinking of large-scale alliances like NATO or even the EU, nor unifications like Germany and Italy, but situations in which a smaller sovereign unit (for whatever definition of sovereignty you want to deploy) asks, successfully or otherwise, to be subsumed into a larger one - like a company being bought by a larger one, in 21st century terms. I have a vague idea that this happened with some of the Anglo-Saxon groups in England around the 4th and 5th century, and possibly with some European states in the 17th and 18th century, but I can't point to any examples.
This has happened in China before. One example, during the Han Dynasty (漢朝), was the Dong'ou Kingdom (東甌國 based around now-Wenzhou). It was under pressure from the Minyue (閩越), so the king asked for and received permission to join the Han Empire. 40,000+ Dong'ou people relocated to the Lujiang Commandary in the Jianghui region. He was given the title Marquis of Guangwu
We also have the case of Hawaii "asking" to join the United States, but that request did not come from the Hawaiian people or the Kingdom of Hawaii. It came from the Republic of Hawaii, established by the American Committee of Safety, which had overthrown the rightful Hawaiian ruler, Queen Liliʻuokalani. Apparently, Texas is not the only state that had been an independent nation :O
司馬遷 Sima Qian. 史記 Records of the Grand Historian. Trans. Burton Watson. Hong Kong: Columbia UP, 1993.