It is unfairness to horse’s nature to move in to the front (8)
I believe the answer is:
inequity
'it is unfairness' is the definition.
(synonyms)
'move in to the front' is the wordplay.
'move' becomes 'equity' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'to the front' says to put letters next to each other.
'equity' after 'in' is 'INEQUITY'.
'to horse's nature to' is the link.
This may not be correct. Some or all of it may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for inequity that I've seen before include "Unfairness, injustice" , "not fair" , "Lack of fairness" , "Prejudice" , "Unjust treatment" .)