ACR X MULTIPLE RWY OP PARALLEL RWYS MADE UNAUTHORIZED TURN WHILE ON RADAR VECTOR FOR VISUAL APCH. FLC SAW WRONG ACFT.

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ATC VECTORED ACR X FOR AN ILS TO RWY 9L. WHEN WE WERE TURNED ON BASE LEG WE WERE ADVISED TO EXPECT A VISUAL TO RWY 9R FOLLOWING A WDB THAT RPTED A 7 KT DROP IN AIRSPD DUE TO WINDSHEAR. WE WERE INSTRUCTED TO MAINTAIN 170 KTS TO INESS BUT WE DID NOT HAVE SUFFICIENT VISIBILITY TO ACCEPT VISUAL CLRNC BEHIND THE OTHER WDB WITHOUT APCH CTLR WE CONTINUED TO INESS AT 170 KTS AND 3000 MSL. INSIDE INESS ATC CLRED US FOR A VISUAL APCH TO RWY 9L FROM 3000 MSL. WITH THE AUTOPLT IN LOC ALT HOLD AND SPD MODE THE CAPT HAD SELECTED 140 KTS ALT GEAR DOWN AND FLAPS 20 DEGS AT INESS. I TOLD ATC WE COULD NOT ACCEPT THE VISUAL TO RWY 9L; SINCE ALL 3 OF US COULD NOT SEE RWY 9L DUE TO HVY RAIN OVER THE RWY; TO WHICH THE CTLR REPLIED; 'OH REALLY; THAT'S NEWS TO ME.' SIMULTANEOUSLY THE STICK SHAKER STARTED; I CALLED IT OUT 'SPD AND PWR.' THE BUG WAS RESET TO 170 KTS AND THE L AUTOTHROTTLE FAILED. AFTER AN ASYMMETRIC THRUST AND LOW SPD RECOVERY WITH NO ALT LOSS WE BEGAN VECTORS S AND W FOR ANOTHER APCH. ON THE DOWNWIND WE WERE TOLD TO FOLLOW A COMPANY WDB FOR A VISUAL TO 9R. WE TURNED TO FOLLOW THE ONLY WDB WE COULD SEE AND ON BASE LEG THE CTLR SAID THAT IS NOT THE WDB YOU'RE TO FOLLOW; CONTINUE WBOUND. WE TURNED BACK W AND THE CTLR SAID FOLLOW THE WDB AT YOUR 12 O'CLOCK AND 3 MI. WE FOLLOWED THAT ACFT TO A SUCCESSFUL APCH AND LNDG ON RWY 9R. OUR DEV FROM ATC'S VECTOR ON BASE WAS NOT INTENTIONAL AND DID NOT RESULT IN AN UNSAFE OP; HOWEVER; I FEEL THE CTLR(S) AT MIA HAVE TOO LITTLE WX INFO. WHEN I SEE THIS SITUATION AND HEAR FROM THEM THAT THEY DON'T EVEN KNOW THAT THEIR RWYS ARE BLANKED IN HVY RAIN. ALSO; THE CTLRS NEED TO BE AWARE THAT LAST MIN RWY CHANGES MUST BE PROGRAMMED INTO OUR COMPUTERS; RETUNED IN OUR FREQ BOXES; IDENTED AURALLY AND THAT THIS ALL TAKES TIME TO ACCOMPLISH PROCEDURALLY AND ACCURATELY SO THAT WE DO NOT LAND ON THE WRONG RWY AND/OR ARPT. SUPPLEMENTAL INFO FROM ACN 214754: AUGMENTED CREW ARRIVED MIAMI FROM MADRID. ALL CREWMEMBERS ON FLT DECK. VECTORED FOR LNDG RWY 9R KMIA. AT 3000 FT MSL INBOUND TO INESS (FAF) ADVISED TO HOLD 170 KTS TO INESS. PRIOR TO INESS ADVISED APCH WOULD NOW BE VISUAL TO RWY 9L. CONTINUING INBOUND COULD SEE HVY RAIN OBSCURING APCH AND THE ONLY RWY VISIBLE WAS RWY 9R. SO ADVISED MIA APCH. ATC THEN TURNED US TO 180 DEG HDG. MAINTAINED 3000 FT. NEVER ACTUALLY CLRED FOR ANY APCH TO THIS POINT. AFTER SOME PERIOD ON 180 DEG HDG; WE WERE VECTORED TO 270 DEGS FOR DOWNWIND. WE WERE ADVISED OUR APCH WOULD NOW BE BEHIND A COMPANY WDB AT 18 MI ON FINAL FOR RWY 9L. SEEING TFC IN CORRECT POS A TURN WAS INITIATED TO VISUALLY FOLLOW AND ATC WAS ADVISED. MIA APCH PICKED US UP IN THE TURN AND ASKED FOR A HARD TURN BACK TO 270 DEGS. WE COMPLIED. ATC ADVISED WE CLOSED TO 3 MI FROM ANOTHER WDB. WE WERE THEN VECTORED FOR ILS 9R APCH AND LNDG. TOO MANY CHANGES BTWN RWYS; TOO MANY CHANGES BTWN IFR AND VFR; AND NO CLRNC FOR APCH.