In succession to throne, oddly, viscount's last (2,3,4)
I believe the answer is:
on the trot
'in succession' is the definition.
(I've seen this before)
'to throne oddly viscount's last' is the wordplay.
'oddly' indicates anagramming the letters (letters in an odd order).
'last' indicates one should take the final letters.
The final letter of 'viscount' 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 "Where Stalinist's likely to pin the blame" , "Moving at a fairly fast pace" , "Continually busy" , "In succession" , "running" .)