AN EMB 120 PIC RPT ON A POTENTIAL CONFLICT THAT CAME ABOUT FROM A DROPPED RWY ASSIGNMENT AFTER MULTIPLE ACFT MISSED APCHS AND MANY HDG CHANGES INTO LAX; CA.

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I WAS THE CAPT AND PLT FLYING ACFT X DEP FAT FOR LAX. AFTER LEAVING DARTS INTXN; WE WERE ON AN ATC VECTOR HEADING 140 DEGS AND DESCENDING TO 4000 FT; AND TOLD TO EXPECT A VISUAL APCH TO RWY 24R FOLLOWING AN ACFT Y AIRBUS AT TWELVE O'CLOCK ON BASE LEG. WE CALLED THE TFC IN SIGHT AND WAS CLRED FOR THE VISUAL APCH TO 24R AND WAS TO MAINTAIN SEPARATION WITH TFC LANDING 25L. THE AIRBUS LATER LOST SIGHT OF THE TFC AND ALSO OVERSHOT THE FINAL FOR 24R. AFTER SEVERAL PROBES BY ATC; THEY FINALLY STATED THAT THEY HAD LOST THE TFC AND DID NOT HAVE THE ARPT IN SIGHT. ATC CANCELLED THE APCH CLRNC AND INSTRUCTED THEM TO TURN RIGHT TO 070 DEGS AND CLIMB TO 2500 FT. BY THIS TIME; WE HAD DESCENDED TO AROUND 3500 FT AND STARTED TO TURN TOWARD THE ARPT IN ORDER TO NOT OVERSHOOT THE FINAL. ATC CANCELLED OUR CLRNC AND INSTRUCTED US TO TURN TO 340 DEGS; MAINTAIN 3500 FT; AND MAINTAIN VISUAL SEPARATION WITH THE AIRBUS AS IT PASSED BELOW US (THIS DID SET OFF OUR TCAS). WE WERE LATER TURNED TO A HEADING OF 070 DEGS AND TOLD TO EXPECT VECTORS FOR AN ILS APCH. SOMEWHERE ON THE DOWNWIND LEG; WHILE RUNNING CHECKLIST; BRIEFING THE FLT ATTENDANT AND THE PAX; AND TALKING TO COMPANY; WE LOST TRACK OF WHAT THE AIRBUS WAS DOING. EVENTUALLY; WE WERE CLRED FOR THE ILS APCH; HEADING 220 DEGS TO INTERCEPT AND MAINTAIN 2500 FT UNTIL ESTABLISHED (ASSIGNED AIRSPEED WAS 170 DEGS). JUST AS SOON AS THE CLRNC WAS ISSUED; WE FLEW THROUGH THE RWY 24R LOCALIZER AND ATC WAS INFORMED RIGHT AWAY. WE WERE TOLD TO CONTINUE THE TURN TO 280 DEGS AND RE INTERCEPT AT 2200 FT OR ABOVE. JUST AS WE WERE ABOUT TO REJOIN THE 24R RWY LOCALIZER; WE GOT A TCAS ALERT WITH TFC AT ONE O'CLOCK AND 400 FT HIGH (I ESTIMATE IT WAS ABOUT 1/4 MI AWAY). IT TURNED OUT TO BE AN F28 DESCENDING ON THE 24R RWY LOCALIZER. WE IMMEDIATELY TURNED LEFT TO A HEADING TO FLY PARALLEL WITH THE TFC. ATC SAW WHAT WAS HAPPENING; CANCELLED F28 APCH CLRNC AND TURNED THEM TO THE NORTH. ATC ASKED US IF WE WERE IN A POS TO MAKE AN APCH FROM OUR PRESENT POS TO RWY 25L. SINCE THE ARPT WAS NOT IN SIGHT AND THE 25L RWY APCH WAS NOT BRIEFED; WE WERE NOT ABLE. WE WERE THEN TURNED LEFT AND GIVEN VECTORS TO A NORMAL ILS APCH TO RWY 25L. DURING A BREAK ON THE GND; I CALLED SOCAL TRACON ON THE PHONE TO FIND OUT WHAT MIGHT HAVE GONE WRONG. I TALKED TO CTLR AND HE TOLD ME THAT THE TAPES WERE GOING TO BE PULLED AND LISTENED TO. HE SAID THAT IT LOOKS LIKE THE ACFT Y AIRBUS WAS ISSUED AN APCH CLRNC TO RWY 25L AND THAT THE CTLR HAD INTENDED TO DO THE SAME FOR US BUT THE NEW RWY ASSIGNMENT WAS NOT ISSUED. AFTER ALL THAT; THIS IS HOW I SEE WHERE THINGS WENT WRONG. AGAIN; I WANT TO SAY THAT I HAVE NOT HAD THE OPPORTUNITY TO LISTEN TO THE TAPES. FIRST; THE AIRBUS TOOK WAY TOO LONG TO ADMIT THAT THEY LOST THEIR TFC AND WERE NOT IN A POS TO MAKE A VISUAL APCH. SECONDLY; DUE TO DIRECTION OF FLT FROM FAT; MOST CREWS ARE EXPECTING SOME KIND OF APCH TO THE NORTH COMPLEX AT LAX UNLESS SPECIFICALLY TOLD OTHERWISE. AFTER BEING VECTORED BEHIND AIRBUS AFTER THE ABORTED VISUAL APCH; I DO NOT RECALL BEING ISSUED A NEW RWY ASSIGNMENT. AND AFTER BEING CLRED FOR THE ILS APCH THE NEXT TIME AROUND; I DO NOT REMEMBER ANY DISTANCE FROM ROMAN OR LIMMA BEING MENTIONED; OR IF THE APCH WAS FOR RWY 24R OR RWY 25L. I FEEL THAT IF I HAD HEARD ONE OF THESE THINGS; IT WOULD HAVE SET OFF A BELL IN MY HEAD THAT WE WERE ON THE WRONG APCH. FINALLY; WHEN WE STATED THAT WE WERE FLYING THROUGH THE LOCALIZER (I CAN NOT REMEMBER IF WE SAID THE RWY 24R LOCALIZER); THIS COULD HAVE BEEN AN INDICATION THAT SOMETHING WAS WRONG BECAUSE OUR RADAR TRACK WOULD HAVE SHOWN US NOT EVEN CLOSE TO THE RWY 25L FINAL.