Obviously this depends heavily on where and how they transfer. While I'm not exactly sure about Danzig/Gdansk, Straßburg and Strasbourg are just the german and french spellings of the same name.

The duchies that used to exist on the border between Germany and Denmark has a similar story - when they changed hands it went from Holsten to Holstein and Slesvig to Schleswig. However some danes still refer to Holstein as Holsten, and I've seen german sources from before they changed hands refer to it as Holstein. I'd say the english nomenclature most often refers to them by the 'official' name from the country currently holding them.

In some cases, the region will maintain a mix of names, reflecting the past and current owners; a good example of this is the Alsace-Lorraine region of France, where Strasbourg is located.

This region is part of the border between France and Germany, and due to the numerous conflicts that have taken place between those countries over the course of history, there is a good mix of French and German in that area. The current accepted name is clearly French; and all the major towns, save for Metz, have distinctly French names. Yet these names have German counterparts, and pronunciation of said towns in the region will differ depending on the person.

So to answer your question, geographical locations are renamed to reflect the transfer between countries, often for the sole reason of further demonstrating the countries control of the area. But within those regions, certain names or aspects will remain in their native tongue to show their origin.

Much like old New York having once been New Amsterdam, cities that have been around long enough have often undergone multiple name changes. Cities founded by Rome have often changed multiple times. Here are a few (more recent) others:

• At the creation of Czechoslovakia after WWI, the city of Pressburg had its name changed to Bratislava (current capital of Slovakia).

• After Ukraine declared independence from the Soviet Union, the preferred English spelling for its capital was changed from the Russian "Kiev" to the Ukrainian "Kyiv".

• Like your Gdansk example, the former German-named cities of Breslau, Posen and Lemburg are now Wroclaw, Poznan and L'viv (which was formerly L'vov).

As an aside, I've always been surprised that so few place names were changed following the American Revolution - we still have many places named after kings or various trappings of the monarchy. I'd be curious if anyone has any insight into this.