What was found
(CAN) DURING CLIMB, FROM FL380 TO FL400, THE RT ENGINE BEGAN TO COMPRESSOR STALL 5-6 TIMES QUICKLY AND BRIEFLY, AS THE CREW, INCREASING POWER AT A SLOW RATE PROPORTIONAL TO THE CLIMB RATE OF THE ACFT. THEIR INDICATED MACH NR WAS APPROXIMATELY .80 AND ENGINES WERE BELOW THE RECOMMENDED CLIMB POWER SETTING. UPON RECOGNITION OF THE COMPRESSOR STALLS, THE CREW ELECTED TO BEGIN SLOWLY DECREASING POWER. BY THE TIME THIS DECISION WAS MADE THE ENGINE WAS IN, ROLL BACK. DUE TO THE CORRELATED POWER DECREASE. WE THEN RECOGNIZED THAT THE RT ENGINE FLAMEDOUT AFTER THE POWER DIPPED BELOW WHERE IT SHOULD HAVE BEEN. THE CREW IMMEDIATELY PUSHED THE NOSE OVER AND ADDED APPROPRIATE RUDDER TRIM TO COUNTERACT THE YAW FORCE. THEY ADVISED ATC THAT THEY WERE DESCENDING AND NEEDED TO START DOWN IMMEDIATELY TO THE LOW 20`S. THEY DECLARED AN EMERGENCY AND ADVISED THAT 12 SOULS WERE ON BOARD WITH 6300 LBS OF FUEL ON BOARD. THE DECISION TO CONTINUE ON OUR ROUTE WAS MADE AS A RESULT OF THE CONVECTIVE WEATHER ON THE RT SIDE OF THE AIRPLANE AND THAT THEY NEEDED TO GO DIRECT TO FARTHER INTERSECTION ON THE FISEL 2 ARRIVAL. AT APPROXIMATELY FL200 THEY BEGAN THE RE-LIGHT PROCEDURE AND THIS PROVED SUCCESSFUL. THEY CLIMBED BACK UP TO FL270 AS THERE WAS STILL WEATHER OFF THE RT OF THE ACFT AND WERE ALREADY BEGINNING THE ARRIVAL PROCEDURE. THE SPEED WAS KEPT AT 300 KNOTS TO PREVENT CORE LOCK AND A POSITIVE ROTATION. THEY LANDED WITH NO FURTHER ISSUES WITH THE RT ENGINE, BUT THEY DID TREAT IT AS IF IT HAD PROBLEM AND MADE SLOW AND MINOR POWER CHANGES. IT SHOULD BE KNOWN THAT THE CONDITIONS AT THE TIME OF THE FAILURE WERE: VMC, OCNL LIGHT TURBULENCE, IGNITERS WERE SELECTED CONT. NO ABNORMAL READINGS WERE SEEN ON THE ENGINE INSTRUMENTS ON ED 1.