Here is my understanding of how a Gamma Stirling works:
Pic 01: The displacement piston ( here blue ) has moved to the extreme left forcing the air in the system into the heating zone ( extreme right ) which takes on heat. Since the system is closed, this causes a rise in pressure which forces the working piston ( here red ) to move outwards ( to the left ). The flywheel gets momentum.
Pic 02: The working piston ( red ) reaches its highest point. The air volume in the system reaches its maximum. Now the displacement piston ( blue ) moves to the right forcing the heated air to move into the cooling zone.
Pic 03: The displacement piston ( blue ) has now moved to the extreme right forcing the air in the system into the cooling zone ( extreme left ). Through the radiating effect of the cooling fins the air can now cool down, producing a pressure drop in the closed system. The power piston ( red, supported by the momentum of the flywheel ) can now move to the right.
Pic 04: The working piston has now moved to the extreme right, the air volume is now at its lowest. The displacement piston now moves to the left forcing the air into the heating zone. The cycle can now repeat.