The evacuations came as Taliban militants seized the district of Qala-e-Zal, just west of the city.
Clashes continue in other parts of the city, as the Taliban try to increase pressure on government troops in both the north and south of the country.
The militants twice entered Kunduz in the past two years before retreating.
BBC South Asia editor Jill McGivering says this is a classic example of surge and retreat which is now common in Afghanistan as the Taliban tries to extend its reach but struggles to retain new ground.
