You might count Free France from 1940–1944:
Capital:
Like Byzantium (that is Roman Empire capital becoming Constantinople), Trier was colonised by the Romans and then made a capital for a time. Speaking of Byzantium, that city had such an honour again, when the Ottomans took Constantinople and renamed it eventually to Istanbul.
The Yuan dynasty of the Mongols and later the Qing dynasty might be described as colonising China and shifting their capitals.
During the War of the Fourth Coalition Königsberg, in East Prussia that first 'had to be' colonised in the previous centuries, was the capital of Prussia for as long as Napoleon's troops occupied Berlin. This is actually a case of a former colony transferring its very name to the German colonisers of Brandenburg as a whole.
Note that this all hinges a bit on definitions. This answer excludes internal colonisation as otherwise also the colonisation of East-Germany by the Federal Republic after 1990 would count as well, as the capital of the internal colonisers was moved permanently from Bonn to Berlin.
But the above refers at maximum to such processes like Russia did with Siberia.
Also note that the officially designated status or "how handled" can be intentionally misleading: Algeria was "part of the French motherland", but I guess these parts of Africa were clearly a colony?
From 1848 until independence, the whole Mediterranean region of Algeria was administered as an integral part of France.
If that example of Algeria wouldn't count, then the premise from the question becomes moot:
When Brazil was elevated to Kingdom in 1815, it became the capital of the United Kingdom of Portugal, Brazil and the Algarves until the return of the Portuguese Royal Family to Lisbon in 1821, but remained as capital of the Kingdom of Brazil.
As the question doesn't narrow the criterion down to "overseas colony in the age of colonialism", we might as well look at the oldest examples. Like the Assyrians, having their name identical to their old capital of Assur, but moving to Calah and Nineveh.
The differences between conquering and colonisation would have to be explored and defined a bit more closely. Otherwise the list of examples might get a little longer. For example including the move of the Russian capital from Moscow to St Petersburg.
A note on the definition proposed as an emphasis on "A country or area under the full or partial political control of another country and occupied by settlers from that country."
Portugal's and many British colonies would have a hard time to really fit into this. Especially for India and Africa, the British colonised without very much settlers compared to the New Zealand or the American colonies. Portugal, unable to afford emigration on such a scale exported even less settlers over time. For the time of the Transfer of the Portuguese Court to Brazil:
When the Portuguese court arrived in Rio de Janeiro on March 7, 1808, Brazil was very sparsely populated, with a little over 3 million inhabitants. Around one-third of the colony’s population consisted of enslaved peoples, most having been captured and shipped from Africa. The indigenous population at the time was of around 800,000 people having been dramatically reduced and isolated during the first 300 years of exploration and colonization. Population density was concentrated along the Atlantic coastline.
Under the definition from the question, let's look again at early Brandenburg-Prussia, when the Hohenzollern inherited the Duchy of Prussia:
(East-)Prussia is the isolated red fleck to the right, controlled from Berlin; further the local Old Prussians, Kashubs, Warmiaks, Poles, Latvians, Lithuanians, Masurians were constantly complemented with ethnic German settlers.
