Back when countries near proximity had access to conventional military gear with equal level of power, all out war made sense, you simply had to destroy the other countries gear to deter them from ever trying that shit again.
but after the development of nuclear weapons, this is no longer a realistic choice. Instead, look at the Russian model:
1) Infiltrate high society of an enemy country decades before saboteur activities. Helps if you have kompromat of sexual nature. Especially, if back home if the truth ever got out, you would be finished. Elites are used to doing whatever the fuck they want and getting away with it so there's little to suspect or take precaution. They are the perfect weak willed asset that can be easily persuaded to serve their own interests which is aligned with the foreign government's.
2) Use your newly acquired assets to spread misinformation through social media. Instigate separatist movements within the enemy country.
3) Wait for the enemy country to react and use it as an excuse to "protect ethnic minorities" or some other bullshit like pipelines.
4) Start arming the rebels, send professional soldiers and latest equipment to make the enemy hurt.
5) Occupy the rebel territory, sign ceasefire, and let UN sit on it's ass for decades until the rest of the world forgets.
6) Repeat.
The American model is a bit different and rarely need to involve the military unless other escalation method fails:
1) Bribe the elite, give out monopolies, lucrative contracts to further American interests at the peril of the victim country's citizens in the name of "Freedom". Target country is cleaned bone dry of resources. If you had oil and lived in the same continent, chances are high that the US is directly responsible for narco-economies in South America.
2) Let human greed corrupt the government inside and out so they are more open to electing a leader serving their own self interests that are aligned with American interests.
3) If somebody doesn't take the silver, they get lead or crash and burn. Make it look like an accident.
4) If all your assets are gone thanks to annoying rebels that took up arms to protect their country from corrupt foreign interests, send in the military to crush them, install someone friendly to the US, and this time give them the latest hardware and weapons aimed at their own citizens.
tldr: How poor and rich empires do empire things slightly differently in robbing the world blind.