You are only allowed to send one person to a given country at a time. For example, this morning's first task was to send someone to San Francisco, USA. If you already had someone on a flight to Gotham, USA, then you would not have San Francisco available on your list. You would need to wait until that flight was returned before sending someone to the correct one.
Also, if you only have a few flights to choose from instead of the full five, try sending one of the people who have a flight available on another task. That should refresh the flight list and give you five to choose from.