How would modern jet fighters engage and defeat prop driven fighters?
Merge

Assuming neither aircraft tried for a head-on guns snapshot, the goal of the jet against the prop will be to go out of plane with the prop post merge and deny the prop a quick guns solution.  By far the easiest way for the jet to go out of plane is to extend and then take the fight vertically.  This means that the jet is in full AB coming into the merge and well above cornering speed, while the prop is coming into the merge at its best corning speed hoping to get a quick snapshot.  As the aircraft approach both pilots tighten down and pass as close as possible to prevent either aircraft from taking angles right after the merge (ie, so that neither aircraft has a turning advantage immediately post merge).

Post Merge

As the aircraft make their first pass, the prop (knowing full well the jet will take the fight vertically) begins a max performance turn hoping to quickly take angles and give itself a quick snapshot against the jet as the jet climbs.  However, the jet, knowing the prop is looking for a radius fight, extends the merge horizontally, at Mach 1+ and then begins a supersonic climb upwards.  Due to the lower thrust to weight ratio of the prop, most of the props potential energy comes in the form of its altitude; however, the jet has liquid potential energy in the form of fuel, and many can accelerate in the vertical, this gives the jet a decided advantage against the prop.  Because the prop cannot climb with the jet, and the jet is now outside of its guns range, the prop must remain at altitude and wait for the jet to attack.

Out of Plane

The jet, moving vertically, has now successfully moved out of plane with the prop rendering the turning radius of the prop moot.  An unchallenged out of plane maneuver is certain victory for the jet.  The jet can now cut the turning circle of the defender from above.  The best hope for the prop is to either pull hard into the jet trying to force an overshoot (and hope that the jet isn't already anticipating this and drawing a wall of lead between this anticipated spot and his current location) or, to wait until the last minute and pull into the vertical with the jet (causing a vertical merge) and hope for a head-on snapshot.  Either way, the jet can quickly disengage vertically again, causing the prop to bleed precious energy with these high G maneuvers, and then reengage when he has the advantage again.  Each time the jet disengages, the prop is left scrambling to regain its depleted energy, and the jet is maneuvering to prevent the prop from ever being able to fully recover.  This means that as the prop dives to gain speed, the jet dives after it, forcing another high G pull from the prop for another overshoot, or, if the prop tries to extend horizontally, the jet attacks again, forcing the prop to use more energy.

Conclusion

The jet will either force a mistake out of the prop and kill it, or the prop will eventually run out of altitude and airspeed and either auger into the ground, or be killed when he doesn't have the energy to force the overshoot, or climb vertically again.  This, of course, ignores things like thrust vectoring, and the flight computers, in aircraft like the super hornet, which give certain Gen 4/5 aircraft supermaneuverability--meaning, post merge, most Gen 4/5 aircraft could just out rate the prop and the fight would likely be over in seconds.