WDB ACFT CLRED FOR ILS APCH AFTER PASSING INTXN WAS RECLRED TO ANOTHER RWY AFTER FMS AND LOC ILS CAPTURE. FLC DISTR BY REPROGRAMMING FMC AND HDG CHANGE DSNDED THROUGH 7000 FT AND WAS RECLRED TO 6000 FT. THEN WHILE DSNDING TO 6000 FT AND BEING DISTR BY TFC; FLC DSNDED THROUGH 6000 FT.

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EXECUTING CIVET 1 ARR TO LAX. CLRED FOR ILS RWY 25L AFTER PASSING ARNES. APCH FLOWN IN AUTOFLT; APCH MODE; LOC AND GS CAPTURED. AT ABOUT (?) 8000 FT; SOCAL INSTRUCTED 'TURN R 10 DEGS; INTERCEPT ILS RWY 24R; DSND TO 7000 FT.' DURING THE CONFUSION OF TRYING TO LOCATE THE RWY 24R APCH PLATE; THE TURN WAS INITIATED BUT THE AUTOPLT FLT DIRECTOR SYS STILL HAD LOC AND GS CAPTURE; COMMANDING A DSCNT ON THE FLT DIRECTORS. WHILE TRYING TO SET UP TO INTERCEPT RWY 24L; DSNDED BELOW CLRED ALT. RECLRED TO 6000 FT; AND WHILE STILL FOLLOWING AUTOPLT FLT DIRECTOR SYS COMMANDS FOR RWY 25L; DSNDED 300 FT BELOW CLRED ALT. IN MY BELIEF; THIS INCIDENT OCCURRED BECAUSE OF THE DELAYED ABANDONMENT OF AUTOFLT COMMANDS UPON RECEIPT OF A DRASTIC CLRNC CHANGE AND THE RESULTANT CONFUSION AND INCREASED WORKLOAD. ALSO; SINCE WE HAD BEEN CLRED FOR THE RWY 25L APCH; OUR MINDSET WAS TO TRACK GLIDE PATH; EVEN THOUGH ONLY CLRED TO INTERCEPT RWY 24R. IN ANY EVENT; THIS EVENT DICTATES THE IMMEDIATE ABANDONMENT OF AUTOFLT UPON RECEIPT OF A CLRNC CHANGE THIS DRASTIC. WITH THE TIME INVOLVED IN REPROGRAMMING THE FMC AND RETUNING THE ILS; AUTOFLT DSCNT ON THE OLD APCH MUST BE CHKED. SUPPLEMENTAL INFO FROM ACN 372334: THE CTLR THEN CLRED US TO MAINTAIN 6000 FT MSL. AT THIS TIME AN ACFT WAS SPOTTED APPROX 5 MI AHEAD. WHILE I WAS DISTR WITH THE AIRPLANE AND THE OTHER PLT REPROGRAMMING THE FMC AND SELECTING THE LOC FREQ; THE ACFT DSNDED TO APPROX 5700 FT MSL WHEN A CLB WAS IMMEDIATELY INITIATED BACK TO 6000 FT MSL. WE WERE THEN CLRED FOR THE APCH WELL ABOVE SGS FOR RWY 24R ILS.