In succession to throne, displaced head of Tudors (2,3,4)
I believe the answer is:
on the trot
'in succession' is the definition.
(I've seen this before)
'to throne displaced head of tudors' is the wordplay.
'displaced' indicates an anagram.
'head of' suggests taking the first letters.
The first letter of 'tudors' is 't'.
'to'+'throne'='tothrone'
'tothrone' anagrammed gives '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" , "in a row?" , "One after another" .)