What immediately comes to mind is the Velvet Divorce, the dissolution of Czechoslovakia several years after the fall of the Berlin Wall. The federation of the two states had largely been propped up by the Communists, and with them removed the two nations found that they had little in common. Czech parties had no real following in Slovakia and vice-versa, and Czechs generally felt that the Slovaks were a drain on the budget while Slovaks resented what they saw as Czechs dictating policy (especially the pseudo-"shock therapy" policies being dictated by Prague at the time). Eventually, during negotiations about how to reorganize the federation, the Slovak national parliament unilaterally declared independence, and while this wasn't a sentiment shared by a majority of either Czechs or Slovaks there also wasn't any real interest in fighting about it.

Norway broke away from union with Sweden in 1905. Although there were a few tense moments, it all transpired peacefully and friendly diplomatic relations were quickly established.

I can think of one example: In the late 19th-early 20th century,Britain gradually granted independance to its white settler colonies: Canada, Australia, New Zealand, and, to some extent, South Africa. This was based on the current British policy of free trade, which meant that they didn't actually need to be in control of a country/colony to derive economic benefits. Therefore it was deemed acceptable to grant independence to countries that were likely to remain friendly to Britain and trade with her, and what better countries than those with a large friendly population. Since then, Aus, NZ and Canada have enjoyed a mostly warm relationship with Britain, even retaining Elizabeth I as monarch. (SA is slightly different due to the fact that the majority of the population weren't of British origin and power was held by a small European elite). I'm on mobile so can't post links, but google "Statute of Westminster", which was a bill allowing the colonies more autonomy, and was arguably the first move toward decolonisation.

Brazil's breakaway from Portugal would be a good example of this with some issues at first between the monarchy and the new nation, but ultimately they were able to set up stable relations given a few years.