ALTDEV BY B737-200 NEAR MIA.

Narrative

DURING CLBOUT FO WAS HAND FLYING THE AIRPLANE. WE WERE BEING VECTORED FOR TFC AND WX (MODERATE RAIN SHOWERS) TO OUR DEP FIX. NUMEROUS HEADING CHANGES WERE ISSUED ALONG WITH A CLB CLRNC FROM 7000 FT TO 16000 FT. LEAVING 14000 FT WE WERE ISSUED A HEADING CHANGE OF 40 DEGS (330 DEGS TO 010 DEGS) AS WE TURNED WE ENTERED THE RAIN SHOWERS (NO TURB). A GA ACFT WAS REFUSING A TURN INTO THE WX IN WHICH SEVERAL PREVIOUS ACFT HAD ENTERED; RADIO COM WITH THIS ACFT AND OTHER ACFT WAS VERY BUSY. AS PNF I GOT CONCERNED WITH THE ALT AND LOCATION OF THE GA ACFT. THE 15000 FT FOR 16000 FT CALLOUT WAS MADE BY BOTH PLTS; WE WERE IN A 25 DEG BANKED TURN AND AT 15700 FT I NOTICED OUR CLB RATE HAD INCREASED TO 4000 FPM. I STATED AGGRESSIVELY; '16000 FT' LEVELOFF BUT THE PUSHOVER TOOK OUR ALT TO 16400 FT. WE AGGRESSIVELY RETURNED TO 16000 FT. NO CONFLICT WAS STATED. WITH THE CONDITIONS; THE PF SHOULD HAVE ENGAGED THE AUTOPLT. ALSO BOTH PLTS SHOULD HAVE CONTINUED TO MONITOR THE LEVELOFF. ATTN WAS DIVERTED FROM MONITORING OUR ACFT TO THE LOCATION OF ANOTHER ACFT.