Those who have prepared and placed themselves and their loved ones in safe places will not find themselves overrun at the last minute. This is not because last minute stragglers are not trying to join them. This is because the stragglers cannot reach them. Imagine the situation. On one side of the Earth the Sun is not setting.
Machines break down.
The telephone lines are jammed, and highways blocked with disabled cars. Those on foot don't last long in the heat. Essentially, all is heat-locked. People will seek a cool spot and wait for whatever comes. On the other side of the Earth perpetual night is reigning.