A B727 FLC HAD A NEAR MISS. ACFT CLBED THROUGH ITS ALT ON ARR TO FLL.

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ON VECTOR TO INTERCEPT MRLIN 4 ARR5 (FLL); WE ENCOUNTERED A NEAR MISS WITH A CPR BIZJET (GULFSTREAM); REQUIRING BOTH ACFT TO TAKE EVASIVE ACTION. GULFSTREAM WAS DEPARTING PBI AND WAS GIVEN TFC INFO (US) BY ZMA. HE INDICATED HE DID NOT SEE US. ATC NOTIFIED US AND WE ALSO COULD NOT SEE TFC (TARGET WAS ON TCASII). OUR CLRNC HAD BEEN TO DSND TO 15000 FT. THE BIZJET SIGHTED US; ATC CLRED HIM TO FL210. DURING THAT XMISSION WE SIGHTED JET AND THEN NOTIFIED ATC. ATC CLRED US TO 11000 FT. WE ACKNOWLEDGED AND CONTINUED DSCNT. VERY SHORTLY ATC NOTIFIED BIZJET HIS TFC (US) WAS CONVERGING AND THAT 'THERE MIGHT BE SOME WAKE TURB.' WE MAINTAINED VISUAL CONTACT AND APPROX 10 SECONDS LATER; TCASII WARNING SOUNDED 'TFC; TFC' WITH IMMEDIATELY 'CLB; CLB.' (NOTE: WE HAD STILL BEEN DSNDING.) WHILE ATTEMPTING TO REVERSE VERT DIRECTION THE BIZJET CROSSED OUR NOSE APPROX 1/2 MI AND APPROX 400 FT BELOW. HE APPEARED TO BE 'PUSHING IT OVER.' THE BIZJET XMITTED TO ATC; 'THAT WAS PRETTY CLOSE.' I XMITTED 'WE CONCUR.' APPROX 15 SECONDS LATER; THE BIZJET ANNOUNCED TO ATC THAT THEY HAD TAKEN EVASIVE ACTION DUE TO A TCASII RA. I TOLD ATC THAT WE HAD ALSO MANEUVERED DUE TO A TCASII RA. ATC ACKNOWLEDGED BOTH WITH 'ROGER.' APPROX 1 MIN WENT BY AND BIZJET AGAIN TOLD ATC ABOUT THE EVASIVE ACTION. THE CTLR BEGAN TO REPLY; STOPPED MID-SENTENCE; AND THEN CLRED THE BIZJET TO ANOTHER ZMA FREQ. I SPOKE WITH AN ATC SUPVR UPON LNDG. HE HAD DEFENDED THE CTLR'S ACTIONS AND THAT THEY WERE ACCORDING TO PROCS. MY COMMENT WAS THAT EVEN THOUGH ALL PARTIES MAY HAVE ACTED PROPERLY; THERE WAS NO REASON TO CLR CONVERGING ACFT THROUGH ONE ANOTHER'S ALT. NO ONE IS IN THAT BIG OF A HURRY. HINDSIGHT; REALIZING THE DEVELOPING SIT; NEITHER PLT SHOULD HAVE CONTINUED WITH CLRNC AND QUESTIONED IT. UNDERSTAND THOUGH; THAT THIS SIT DEVELOPED EXTREMELY QUICK. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: THE RPTR STATED THE CTLR ISSUED TFC TO BOTH ACFT BUT EACH WAS UNABLE TO ACQUIRE THE OTHER. THE CTLR ADVISED THE CLBING ACFT THAT IT MIGHT BE CLOSE WITH THE POSSIBILITY OF WAKE TURB. AT ABOUT THIS TIME THE DSNDING ACFT RECEIVED A TCASII RA TO CLB. DURING THE CLB ESCAPE MANEUVER THE RPTR OBSERVED THE INTRUDING ACFT IN A 'PUSHOVER' MANEUVER. BOTH ACFT TWICE NOTIFIED ARTCC THAT 'THAT WAS PRETTY CLOSE' AND THAT BOTH HAD TAKEN EVASIVE ACTION WITH NO RESPONSE. WHEN ARTCC STARTED TO REPLY THE RPTR FELT THAT THE XMISSION WAS INTERRUPTED AND THE OTHER ACFT WAS ASSIGNED ANOTHER FREQ. A POSTFLT TELEPHONE CALL TO THE ARTCC FACILITY FOUND THE SUPVR VERY DEFENSIVE OF THE CTLR ACTIONS. THE RPTR FILED A RPT WITH HIS COMPANY AND WAS INFORMED THAT THE CTLR HAD BEEN REMOVED FROM HIS POS AND HAD RECEIVED SOME ADDITIONAL TRAINING.