I too have worked with the Red Cross and it is not their responsibility to pay for flights when military personnel return to for a family emergency. It is their responsibility to transmit the emergency communications between the family and military, get the supporting documentation about the emergency for the military and help make the flight plans. The military and the service person themselves are responsible for the payment.
And I know for a fact that all disaster assistance provided by the Red Cross is free to victims so I highly doubt the veracity of the story about charging victims for coffee.
I find it best to call the organization to ask for clarification before I say negative things about it in a public forum.