London had a population of 20,000 Africans by the 1700s I may add, not to mention established communities of Bengali lascars, Chinese sailors and large numbers of Jews, Irish and Huguenot refugees - the last of which its said 50% of 'native' Londoners are derived from (and another, overlapping third of Londoners have Irish ancestry). This was the beginnings of empire - when the first Emglishman set foot in India already there were 3000 Indians living in Britain.
Basically what you are seeing are
the rise and fall of multiculturalist waves - Roman, Saxon and medieval London were s
trongly foreign born, yet with intermittent waves of homogeneity as the children of these emigrants were assimilated. The apogee of assimilation was in the 1920s, where although many Londoners had foreign ancestry, much of it was lost through the generations and barely survived even in name.
This tradition continues, many British born emigrant children already identify with British within one generation - the last leader of the main right wing party, who was anti -immigration was recently 'exposed' as the son of Romanian immigrants who had changed their names. Also the country has the highest intermarriage rates in the West, with 'mixed race' the fastest growing ethnicity. For example although the Caribbean community is huge, its actually endangered, as more than half its members are in mixed relationships. This is the same for continental Europeans and East Asians, and accounts for a third of South Asians.
So in other words London's next generation will be less mixed than the current one - and unlike the US, many people do not identify as Irish/ Polish American etc even within one generation let alone several.
This rise and fall is echoed across Europe,
the Roman Empire mixed things up greatly for example, Italy was of course much much more diverse than it is today with every European peoples represented along with Middle Easterners, North Africans, sub-Saharan Africans, Central Asians, and even East Asian slaves and traders. Even after the Roman Empire collapsed it remained very mixed - bear in mind there have been several African Popes since.
Likewise with Asia, the great cities of the Silk Road, and the capitals of Beijing, Xian and Nanjing were vast magnets for global trade, religious pilgrims and conquered peoples, such as slaves, servants and soldiers. Not only did they number all the East Asian peoples, but drew travellers from Central Asia, the Middle East and even Europeans such as Greeks, Romans, Gipsies and Jews having communities there too.
Of course the multiculturalism died with assimilation and intermarriage and this is echoed throughout the world. And still is happening today in a cycle of rise and fall.
Take India for a recent example under the
Raj, - one third of the English soldiers stationed there took Indian wives and had children, and this English community still survives. Staunch Protestants they retain their English language, names and traditions, and hold marriages only within their community - they now number 800,000 around the world. In India they are traditionally employed on the railways and are known as the Anglo-Indians. Despite their community protectionism this is still the modern day result:
Jacob and Isabella Thomas, with sons Kevin and Shaun (left), Leon Nathaniel (middle), the Gomes family, Calcutta (right):

Sangamam Kochi 2005,
www.flickr.com,
http://newsimg.bbc.co.uk
20th Century Anglo-Indians - note the British fashions transported straight from London.
Matthew and Coleen Demeder:
Celeste33,
http://www.flickr.com/photos/celeste33/40640521,
The Martin Family:

Shazzbazz draws the Line!,
http://www.flickr.com/photos/39117859@N00/2741178972
Russell and Enid Fonseca and family:
http://www.sumgenius.com.au
*Of course India is possibly the worlds most multicultural country, vying with certain African countries and Papua New Guinea. 600 languages, thousands of dialects, hundreds of ethncities and thousands of castes.