Every tile has a cultural culture value - even if that value is zero, and cities can assert their culture on the world and push out their boundaries in a variety of ways.
'Mechanics'
Deranged Duck's Culture Mechanics Disassembled explains this in detail (but respectfully, it's not well laid out and needs both images and shorter paragraphs with paragraph titles).
'Cultural Assault' games
Some interesting succession games re. 'cultural extermination' are;
These above games are different from Cultural Victory games where you need three cities to hit Legendary status to win. See the Peacemonger Strategy section in the War Academy for ideas on winning by Culture.
Pressing Against or otherwise Asserting Cultural Pressure
The shorter answer to your question would be to combat cultural encroachment with culture of your own with tactics that might include:
Selecting a leader with the Creative trait,
Building Wonders (in most cases) and founding Shrines and Academies add ... the Sistine Chapel adds a further bonus as one of several -oriented Wonders. Note also that early-built Wonders also may double their cultural output.
Constructing culture-generating buildings (Obelisks/Monuments {also Stonehenge}, Temples, Libraries, etc. ... anything that adds points to the city),
Turning up the slider (requires Drama - often used in Cultural victories, dealing with rampant unhappiness, or trying to grab land towards the conclusion of a Domination win), or 'building Culture' (the symbol on a city's build list),
Some Corporations when in a city add (Corporations are a Beyond the Sword expansion pack feature),
Running Specialist Artists, and using Great Artists
... are some ways of lifting the cultural output of cities (I'm bound to have forgotton some! ).
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