How do messages get from the destination airport to a pilot? A captain waiting in her airliner for take-off at an airport might inform her passengers:


  We'll be waiting another ten minutes before we're able to take off, as traffic is very busy at [destination], and we have to arrive in accordance with the landing slot we've been given.


How did that information reach the pilot, and what route did it take?

The answer I am looking for might be something like: the slot controller allocates the slot in the local slot control system database. This is communicated via a dedicated digital radio satellite system to ATC at the departure airport. ATC informs the local despatch controller by telephone or short-range radio. Finally, a runner is handed a slip of paper with the details and personally puts it into the pilot's hands.
When the crew got their clearance for the trip from Clearance Delivery over the radio in the flight deck, it would've included a departure time window with the delay, and they may have been told by the CD controller why, or may have figured it out for themselves because it's a routine occurrence for the airport they were headed to (like Chicago).

Or, they were told prior to that during the planning brief with Operations. All the planning and filing is done by specialists in the airline's Operations department. The crew just shows up at Operations and picks up a package with everything prepared; the route, weather, provisional loading, fuel, etc. Going to an airport with a reservation protocol in place, they would have been briefed on that by Operations. That is all provisional however, the final routing, loading, fuel etc is determined when getting ready at the gate, and also when they get their ATC clearance by calling Clearance Delivery over the VHF radio before programming the Flight Management Computer.