Runs eastern diplomatic mission in exile (10)
I believe the answer is:
relegation
'exile' is the definition.
(thesaurus)
'runs eastern diplomatic mission' is the wordplay.
'runs' becomes 'r' (cricket abbreviation).
'eastern' becomes 'e'.
'diplomatic mission' becomes 'legation' (legation is a kind of diplomatic mission).
'r'+'e'+'legation'='RELEGATION'
'in' is the link.
(Other definitions for relegation that I've seen before include "that's going down" , "Transference to a lower league" , "Demotion, to a lower league" , "Lowering of status" .)