The phrase 'assimilate themselves' is correct and usable in written English.
You can use it when you want to express that a group of people have integrated or blended into a different culture or environment to the point where they are almost indistinguishable. For example, "The immigrants worked hard to assimilate themselves into their new society."
It draws its social norms from the mores of the Anglo-Saxon diaspora — and it expects new arrivals to assimilate themselves to these norms, and quickly.
Twin Peaks: Fire Walk With Me (1992) If anyone could assimilate themselves into David Lynch's Twin Peaks universe with ease, it was David Bowie.
The full name, then its derivation: her parents were Russian immigrants who knocked the Slo off the front of Slobodkin to assimilate themselves into the American dream.
"I think when an immigrant community wants to assimilate themselves into the political process," he said, "they latch onto whatever they can find".
"I didn't want to do a movie about illegal immigrants or mail-order brides, which are very trite subjects now, so I said, why not do a story that's a valentine to New York, about a group of people who are able to assimilate themselves without much trouble, because that exists also?
In order to re-ascend and return to the "original, true self", practitioners of Falun Gong are supposed to assimilate themselves to the qualities of truthfulness, compassion and tolerance, let go of "attachments and desires" and suffer to repay karma.
("Cats," says Cornelius Appin, the guest in question, "those wonderful creatures which have assimilated themselves so marvellously with our civilization while retaining all their highly developed feral instincts").
The Caliphate that came to be established was an Islamic state ruled by Arabs, but very soon non-Arabs who had assimilated themselves to the new situation began to participate in the affairs of the Muslim community.
Kennedy was one of more than 100 undercover officers who, over the previous four decades, had transformed themselves into fake campaigners for years at a time, assimilating themselves into political groups and hoovering up information about protests that they had helped to organise.
As the Chinese government began to re-assimilate themselves into the global community in the late 19th to early 20th century, it shifted away from conventional Confucian ideals and women's role in society changed as well.
After acquiring their own recording studio, the duo immediately started working on new material and assimilated themselves with music recording and studio equipment, as they sought to become more ambitious artists and less dependent on other producers.
