American Retirement in Serbia or Russia

I am planning to permanently relocate overseas when I retire. Many Americans that retire in other countries stay there for years without being a full citizen in that country. How do they do this? There are "residence visas/permits" that are temporary lasting a year or so or "permanent". I realize the laws for long term residence are different for every country. If you have a temporary residence visa do most countries let you renew it an infinite number of times? Do you have to return to your native country several times due to your status? Some of the countries I am interested in are Serbia and Russia.