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Having done some research on Urban Renewal, it seems that is a phenomenon that occurs for the same reasons in African and Asian nations as it does in Western nations - in that it occurs when it needs to in order for an area of a city to be rebuilt or repurposed. One of the best papers I found concerning Urban Renewal is "Neighborhood Regeneration in Beijing" on the McGill University School of Architecture website.
Experts present at the first International Seminar on Urban Renewal, held in Den Haag in August 1958, agreed that the main purpose of urban renewal is to deliberately change the urban environment and to inject new vitality through planned adjustment of existing areas to respond to present and future requirements for urban living and working (Miller, 1959).
"Neighborhood Regeneration in Beijing" - McGill University
As you quite rightly mentioned, in the late 19th century London was the focus of one of the first instances of state involvement in Urban Renewal (previously it was a process that was privately initiated, a notable example being during the industrial revolution) in order to combat the unhealthy living conditions in working class neighborhoods. (As an unrelated side note, the Boer War illustrated to the British Government that there was a serious issue with public health in some areas as "in some towns as many as nine out of ten recruits for the army were rejected because they were so unfit" - BBC).
This principle of urban renewal, public health and safety, can be seen as a driving force behind the process of urban renewal in Hong Kong that took place in 1954. The renewal of the Shek Kip Mei district was initiated by Governor Alexander Grantham after a huge fire in a slum saw some 53,000 people become homeless in 1953. As a result the Shek Kip Mei Low-cost Housing Estate was constructed.
The urban renewal that took place in Singapore in the 1960's was also as a slum clearance measure as some 25% of the cities then population was thought to be living in as little as 1% of the available land. This was a result of the huge amount of damage done during WW2.
The importance of both world wars in terms of urban renewal cannot also be understated. Huge amounts of damage were done to cities across Europe as well as cities in China, Japan and northern African nations.
Rather than write my own summary, the following quote from "Neighborhood Regeneration in Beijing" seems to fit perfectly with what I have found so far, and with what I was taught when studying Urban Geography while at college.
In summary, it can be observed that, both in developing and developed countries, the evolution of policies regarding urban renewal followed a similar pattern, gradually evolving from a demolition and reconstruction approach to a softer, more socially-oriented approach, which concentrates on the renovation of existing structures. The following section summarizes the possible approaches to urban renewal in residential areas, based on this brief review of urban renewal policies around the world.
"Neighborhood Regeneration in Beijing" - McGill University