Animals are able to go without food and water for several hours. It is therefore preferable to give them their last proper meal 12 hours before placing them in their travel cage.
However, before dropping off your pet at the airline company, you may give them a small quantity of water and food in the dishes clipped onto the door of the travel cage. The less the animal will have to digest during the trip, the lower the risk of soiling the cage will be. This will make their trip more pleasant.
Animals are only fed by the airline company if they remain in transit, but not in the case of a direct flight.