Mumbai, India's financial capital (an NEE city), has grown rapidly to around 21 million people in its wider metropolitan area, and its growth shows how several causes combine.
The main trigger was rural-urban migration driven by strong pull factors. Mumbai offers jobs in finance, the Bollywood film industry, the port, textiles and a vast informal sector, together with better access to schools, hospitals, water and electricity than rural India. Powerful push factors reinforce this: rural Maharashtra and neighbouring states suffer poverty, low and unreliable farm incomes, debt, drought and seasonal unemployment, so families move to Mumbai hoping for a better life.
However, migration is not the only cause. Natural increase is now very important: most migrants are young adults, so birth rates remain high while healthcare lowers death rates, meaning the resident population grows on its own. Because Mumbai's population is already huge, this natural increase now adds more people each year than migration does.
Overall, I would assess that rural-urban migration was the most important cause of Mumbai's initial rapid growth, but natural increase has become the dominant cause of its continued growth. The causes are therefore linked rather than separate — migration created a large, youthful population whose natural increase now sustains rapid urbanisation. This makes the growth difficult to slow simply by reducing migration.