B767 FLC DECLARED AN EMER AND DIVERTED TO LAND FOR AN ILL PAX RESULTING IN DUMPING FUEL AND AN OVERWT LNDG. THE FLT WAS DELAYED AND REDISPATCHED SHORT OF ORIGINAL DEST DUE TO FLC DUTY TIME LIMITS.

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DURING CRUISE FLT AT FL350 WE WERE ADVISED OF ONE OF THE PAX (AN XA YR OLD WOMAN) WAS COMPLAINING OF NAUSEA AND LOW BLOOD PRESSURE. 2 WELL QUALIFIED SURGEONS VOLUNTEERED TO ASSIST HER AND DISPATCH WAS ADVISED. DR XXX ADVISED THAT THIS PAX WAS IN CRITICAL CONDITION AND ADVISED THAT WE SHOULD GET HER TO THE HOSPITAL ASAP; THE YYY PHYSICIAN CONCURRED. THE BEST DIVERSION STATION WAS DETERMINED TO BE LIMA; PERU AND A CLRNC WAS OBTAINED AND THE ETA WAS 1 HR 20 MINS. THE CTR TANK FUEL WAS DUMPED AND THE DECISION WAS MADE TO LAND IN LIMA AT A GROSS WT OF 330000 LBS. NO ONE THOUGHT TO GIVE THE SECOND MEDICAL KIT TO THE DOCTORS SINCE IT IS THERE PRIMARILY FOR RE-DISPATCH AND WE WERE BUSY WITH DIVERSION ACTIVITIES. IF THE FLT ATTENDANTS HAD KNOWN ABOUT IT; THEY WOULD HAVE REQUESTED IT BECAUSE THE DOCTORS' MAIN CONCERN FOR THE PAX WAS THE LACK OF IV FLUIDS. AFTER LNDG IN LIMA; THE PAX WAS REVIVED AND THERE WAS A DELAY IN HER DEPLANING DUE TO HER REFUSAL TO LEAVE. I WAS ABLE TO CONVINCE HER TO LEAVE WITHOUT CALLING SECURITY. LIMA FLT SUPPORT COMPLETED THE OVERWT LNDG INSPECTION; FUELING; AND MAINT CABIN CLEANUP WHERE THE SICK PAX WAS LYING AFTER SOME DELAY DUE TO THEIR FLT'S DEP NEEDS. AFTER DIVERSION TO LIMA; PERU; AND TAKING CARE OF A SICK PAX; WE WERE ONLY ABLE TO COMPLETE THE NECESSARY AIRPLANE INSPECTIONS; FUELING AND CABIN CLEAN-UP AT XA20Z. WE DECIDED THAT WE COULD ONLY CONTINUE TO MIAMI DUE TO CONCERNS OVER PLT FATIGUE AND NOT WANTING TO VIOLATE FAA DUTY TIME LIMITATIONS. WE WERE DISPATCHED TO AND STOPPED IN MIAMI. ANOTHER FLC CONTINUED THE FLT TO CHICAGO.