Throne to supply Debrett's ultimate in succession (2,3,4)
I believe the answer is:
on the trot
'in succession' is the definition.
(I've seen this before)
'throne to supply debrett's ultimate' is the wordplay.
'supply' is an anagram indicator (supply of letters to make).
'ultimate' suggests the final letters.
The last letter of 'debrett' is 't'.
'throne'+'to'='throneto'
'throneto' is an anagram of 'onthetro'.
'onthetro'+'t'='ON THE TROT'
(Other definitions for on the trot that I've seen before include "without a break" , "Continually busy" , "Informally busy like slow run" , "running" , "In succession" .)