Basically how did China with Mongolia.
For example, this could be done randomly after the conquest of your capital or a border city of yours. Like, you've been conquered and the turn after a message displays : "your city absorbed Mongolia's culture !", your city is given back to you and the war ends for at least 10 turns or so. Of course, it is unsure it would be any beneficial to you, considering the possible loss of population. But it could be a thing, that may be there randomly for everyone.
Or, provided by a wonder (giving a chance) or a policy card, civilization trait or whatever.
Or, it could be based on culture : if you are a vast empire like China with lots of theater squares, and conqueror is basically a refined barbarian, but a barbarian still (pastoralists), the weight of your culture compared to the low culture of your conqueror could convert your city back, more or less quickly depending on the diffence.
Unlike earlier Civ renditions, we don't really have 'culture wars' in Civ VI - you can't flip a city with culture alone, except sort of with Eleanor, and 'culture pressure' really isn't a Thing.
That's not saying it wouldn't be a good idea.
China, by the way, didn't just do it with the Mongols, but also with the Manchus and virtually eve4ry other "Northern Barbarian" that wandered south. On the other hand, although it can be argued that Greece had a higher over-all 'Culture' than Rome, Rome didn't turn 'Greek', it turned into a Hybrid of Greco-Roman culture and then, in Byzantium and the with addition of Christianity, into something almost entirely new and different. Likewise, Gaul had a pretty advanced and coherent culture that was distinctly Un-Roman, but after the Roman conquest it was virtually eliminated: only a few place names and city foundations remain, and modern France conspicuously counts the Romans as their antecedents and ignores any possible Celtic/Gallic elements.
So, "Culture Pressure", I think, will have to come from a combination of things:
1. Pure Culture: which we already measure in the game.
2. Culture times Population: China's huge advantage over the Northern Barbarians was simple population pressure: there were a thousand or more Chinese for every Barbarian/Mongol/Manchu, AND they had a culture and lifestyle that was more appropriate for the physical area of China than the pastoral lifestyle and culture of the northern plains. Result was that while China rarely adopted elements of the 'Barbarian' culture (mounted archery and knight-like lancers by the Tang Dynasty, for instance) the Chinese literature, city life, luxuries, et al. were adopted wholesale by the Invaders, and they discovered that they wound up also adopting Chinese Culture with them, which sees into:
3. Technology. Technology frequently equals Comfort, or at least a more comfortable lifestyle for the Leaders, and so more advanced technology will be adopted in a heartbeat. Then the adopters discover that Technology is accompanied by Culture elements from the society that had the technology, and the assimilation or Hybridization of the conqueror's culture begins. IF you want Chinese porcelains, silks, theater, tea, et al (most of which also represent Technological factors like specialized Porcelain Kilns) then you cannot keep a pastoral lifestyle, because you cannot cultivate a silkworm or throw a porcelain pot from horseback.
Finally, there is the Anti-Culture aspect. If, like the Mongols and some of their successors further west from China, you massacre all the natives of the cities, you avoid all the problems of 2 above: there is no population pressure left. See Timur the Lame's "mountains of skulls" or the Mongol massacres in the Islamic cities of the Middle East (and Bye, Bye Baghdad). Another way to avoid Culture Pollution and Assimilation is to not leave your chosen physical surroundings. The Mongols that stayed behind in Mongolia are still there, still Mongolian. The Scythians of what is now Ukraine remained firmly pastoral nomads despite the Greek city states scattered along the Black Sea coast from modern Romania to the mouth of the Don River, because the Scythians were in their Home Element. They traded with the Greeks, but there was no reason to adopt a lifestyle/culture that was less adopted to their environment than the one they had already developed over the previous 500 years.
All in all, I like the idea of 'Culture Was' assimilation or adoption, but to get all the ramifications right, it may have to wait for Civ VII.