The original construction began in 1708 when the then Duke of Marlborough was leased the land by Queen Anne. The house passed through several hands before the crown again took ownership in 1817 in order for Princess Charlotte to take up residence there. Several other royals lived there in the next 30 years before it became the first site of the Museum of Manufactures. However, the museum was only located at Marlborough House from May to September of 1952. Afterward, it moved up the street to Somerset House before moving again to its current location in South Kensington.

The reasons behind these moves aren't definite, but some sources say that it was because it was thought that the museum would get more patrons in Kensignton (and it did.) From 1853-1861the building was used by the National Art Training School. Later, it again became housing for many more royals including King Edward VII and Queen Victoria. In 1965, Marlborough House was leased by Queen Elizabeth II to the Commonwealth Secretariat and they remain in use of it today. This historic building passed through quite a few hands and was the home of many royals over the last three centuries. It was also the birthplace of what is one of the largest museums in the world: The Victoria and Albert Museum. 

Sources:

https://en.wikipedia.org/wiki/Marlborough_House

https://en.wikipedia.org/wiki/Victoria_and_Albert_Museum 

https://www.vam.ac.uk/blog/caring-for-our-collections/first-years-south-...