Which is why I asked you what elements distinguish between “colonization” and “migration”. While I’m not an expert in history, it seems to me the first Europeans to build colonies in North America didn’t need to subvert any existing power structures. When they had disagreements with the natives, they had gunpowder.
That's not how it happened.
The Spanish worked with tribes that opposed the Aztec in their conquest of Mexico. They worked with neighboring tribes and with divisions within Inca society to overthrow the Inca. Mayan society was already pretty well collapsed, there was no functional government to challenge the Spanish in that area.
Further north, the English and French colonists routinely took sides in wars and conflicts between tribes, pretty much from the day they landed.
The British conquest of India would be a good example of how alien societies could overthrow us.
Start with a trading post. Make friends. Guard the trading post. Guard some of the friends because they help make the trading post work, sell them weapons while you are at it. Guard some of the supply routes and more of the area around the trading post. Make more friends. Help friends in conflicts, because you'll win and that makes the trading post more successful. Let some of the friends hire some of your officers to train their men for fights with people who are not your friends, train them to fight with the advanced weaponry you give them. Let them hire out entire units from your guards if they want, they'll certainly win conflicts that way.
Pretty soon, it gets to the point that the only way to gain power and influence is to be friends with you, and to make use of the military/security services you offer.
Just keep going like that.
By the time your friends don't want to be friends anymore, you already have significant military forces in the area, and control of the better parts of their military. Just look at the events leading up to the 1857 War. The British were not the only ones to follow that example, the the Russians did it throughout central Asia, the French did it in Africa and North America.
One could argue that the U.S. and U.S.S.R did it during the Cold War, and that the U.S. is still doing it.