As a Culture-bound benevolent dictator, an auto-king can only successfully rule people of own culture. Populations of different cultures affect each other, especially if people of these cultures engage in war or trade. Culture affects how people react to choices made by their auto-king, like purging humans, starting wars, losing or conquering worlds.
Cultures have opinions of each other, which are affected by their actions. Auto-king is considered a responsibility of its culture, so blame for its actions is also placed on the people. If a culture hates other culture, they will perform against them better in a war or covert actions (but not the reverse), and if they like other culture, they will both spread culture and get other culture spread better.
Culture can spread remotely, via creation of popular works of art, or directly, via people of different culture existing in same place. Typically this happens as a result of allowing foreign traders to visit systems
As people in a world gain sympathy for foreign culture, they will get increasingly similar reactions to actions of auto-king as foreign citizens. If spread passes certain treshold, culture changes to foreign, and planet gets severe happiness and efficiency maluses. At this point, it can either be held in check with martial law, possibly reeducated (difficult) or returned to culture owner, which will make the culture grateful.