Old hand ran after horse doctor on horseback (7)
I believe the answer is:
veteran
'old hand' is the definition.
(synonyms)
'doctor on horseback' is the wordplay.
'doctor' becomes 'vet' (vetting is a kind of doctoring).
'on horseback' becomes 'eran' (I can't explain this - if you can you should give a lot more credence to this answer).
'vet'+'eran'='VETERAN'
'ran after horse' acts as a link.
I am not very happy about this link. Some or all of it may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for veteran that I've seen before include "Past master" , "Old; experienced" , "Old stager" , "Person with long experience like old soldier" , "Warhorse" .)