During the Napoleonic Wars Prince regent John VI feared being deposed by the invading French forces, and with British assistance was able to successfully flee with the Royal Court in 1807 to re-establish rule from Rio de Janeiro, Brazil. It remained the seat of government for 13 years until re-establishing itself in Lisbon.

As far as I know, that is the only case of a capital displacing outside of the continent. Though movement within a nation has been relatively commonplace throughout history. Military defeats and lost territory has been a relatively common cause for movement of a capital; prime examples being the Song Dynasty in China fleeing to Hangzhou, or the (Western) Roman capital being transferred from Milan to Ravenna in the false belief it would be more easily defended due to terrain and better ocean access, even though it has a history of being captured multiple times.

In other instances, such as the establishment of Charlemagne's court at Aachen, it was the previous home of seasonal royal residence which the leader took a liking to.

As for the establishment of a capital, Washington D.C. was created as an act of compromise.

During the American War for Independence massive debt was incurred by both the states and the central government in order to sustain the war effort. With the ratification of the Constitution, their was much debate over how to go about handling this debt in order to bolster the fledgling American economy.

Then Secretary of the Treasury Hamilton was left with the task of establishing faith in the new found government, which in turn would encourage investment. In order to do this, he had to figure a way to pay off the crippling national debt, most importantly to citizens who had loaned funds, and tying that debt to the federal government.

In short, in order to garner the needed support for Hamilton's plan he had to get major players (predominantly the Agrarian movement) on board. To this end he had to work out a deal with Hamilton and Madison, who were strongly opposed. Through a series of bargains, they eventually settled on the assumption of all state debt by the federal government, in return for the establishment of the national capital on land donated by Virginia and Maryland.

This allowed Hamilton to carry out his plans (which worked quite well) and receive charter for the Bank of the United States. And appeased Southern concerns over centralization of the government in the Northeast by moving the capital farther South, instead of its current home in New York City.

Sources:
Flight of the Emperor by Laurentino Gomes
A History of East Asia by Charles Holcombe
Fall and Decline of the Roman Urban Mind by Svante Fischer, Hans Lejdegård and Helena Victor
The Founding Brothers by Joseph Ellis