The first plane has already left, and two more trips are scheduled overnight.
Britons will have to make their own way to the airfield, navigating potential checkpoints and fighting, as no escorts are being provided.
Priority is given to families with children, the elderly, and people with medical conditions.
There are concerns about the 72-hour ceasefire, which began on Monday.
The Foreign Secretary has said that it is impossible to predict how long this opportunity will last.