B737-300 FLT CREW WHILE TAXIING FOR TKOF HAS A TXWY INCURSION.

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WE ASKED RAMP TWR FOR PUSHBACK. RAMP TWR ASKED; 'CAN YOU DO A QUICK PUSHBACK AND TAXI OUT; DUE TO A FLT COMING IN; IN A BIT; WITH MEDICAL EMER?!' I SAID; 'YES.' HE CLRED US TO PUSH. WE DID ALL CHKLISTS AND ASKED FOR TAXI. HE CLRED US TO TAXI; AND TAXI QUICKLY (THE FLT WITH THE MEDICAL EMER JUST LANDED AND WOULD BE TURNING THAT WAY SHORTLY). I TOLD RAMP TWR I WOULD SWITCH TO GND AT SPOT X. RAMP TWR TOLD ME SPECIFICALLY 'DO NOT SWITCH TO GND CTL UNTIL AFTER SPOT X AND THE R TURN ON TXWY ASAP.' I IMMEDIATELY SWITCHED TO GND (DIDN'T MISS A HALF SECOND) AS WE TURNED R ON TXWY AND CONTACTED GND. GND CLRED US VIA TXWY TO RWY AND THEN CHEWED US OUT. WE TOLD HIM WE DID EXACTLY WHAT RAMP TWR TOLD US TO DO BECAUSE OF THE MEDICAL EMER. HE SAID I HAVE MY 'SPACE' AND HE HAS HIS 'SPACE' AND HE CANNOT CLR YOU IN MY AREA (OR SOMETHING TO THAT EFFECT). HE ALSO SAID SOME COMMENT ABOUT 'I HAVE TOLD YOU BEFORE' (I DON'T KNOW WHO HE WAS REFERRING TO; BUT IT WAS NOT US). I TOLD HIM WE DID EXACTLY LIKE WE WERE TOLD BY RAMP TWR; BUT WE WERE SORRY. I WISH RAMP TWR AND GND ATC WOULD GET ON THE SAME PAGE. WE GOT CAUGHT BTWN 2 'GND' TURF WARS.