If you mean what mechanically moves, there's very little actually, other than in supersonic aircraft. Conventional jet engines (other than ramjets and scramjets) don't work with supersonic airflows into the engine, so there are various methods of slowing down the airflow, usually by a static shockwave generator type thing, like the little bar in the front of the F-16 intake, becuase airflow behind supersonic shockwaves is subsonic. There are a few aircraft specific things, like the drooping intakes on the F-15 to direct air into the intakes at high AOAs, or the alternate intakes on the top of the MiG-29 so they don't ingest debris. But the only thing that will change in a jet engine throughout a flight is the engine RPM, nozzle deflection, and maybe the size of the opening of the fuel valves, but jet engines usually don't have too many parts, and in the case of ramjets and scramjets, they have
no moving parts, which is really amazing.

If you mean what is happening in the engine to produce thrust, just remember "suck, squeeze, bang, blow", which usually works for most power production cycles. The engine pulls air in from the outside with a fan or series of fans on the front. The air is then passed through a series of compressors, then a portion goes through the burners where fuel is added and combusted, which heats it and the rest of the air, which adds a large amount of energy to the airflow. It is then forced out the back where it expands, and forces itself out the nozzle where all that extra energy pushes the jet engine through the air, and if the engine is attached to an airplane, the airplane goes with it. This physical processes of this cycle don't change throughout a flight unless the airflow into the engine is to little to sustain engine RPM, which causes a flameout. Other than that, the only thing that changes throughout the flight is the energies imparted on the air and the engine, and the flowrates and velocities of the air flow throughout the enigine.
If that's not what you meant, than what the heck do you mean?
