This process would most likely take place in MEDCs, because LEDCs are still in the process of urbanization.
Re-urbanization is when people move into areas in the urban zone's which have previously been abandoned, which usually occurs after a regeneration scheme is used for that area.
Example: London Docklands
In the 1980's the Docklands development corporation took the task of regenerating of London Docklands. At the end, they were able to Physically, Socially, Economically and environmentally regenerate the docklands.
One of the reasons why London would chose to regenerate the Docklands instead of suburbanizing is because that London is already too huge of a city, UK is a small island, if London grows anymore, it will be impossible to manage.
Docklands is already in the inner city, transportation is in place, all there is to be done is to build up the buildings around the Docklands. Many old buildings can be re built as well.
For example, old warehouses could be remodeled to become student flats.
It also increases the job opportunity in the inner city, so people don't need to move out to the suburbs to find decent jobs.
All in All, it is good for cities to re-urbanize the area's in the inner city, in order to maintain the size of the city, beautify the city, keep the economy stable in the city.