Flier may get move, right to the south of European capital (10)
I believe the answer is:
budgerigar
'flier' is the definition.
(budgerigar is a kind of birdanimal)
'move right to the south of european capital' is the wordplay.
'move' becomes 'budge' (to budge is to move slightly).
'right' becomes 'r' (common abbreviation).
'to the south of' means one lot of letters go next to another (in a down clue, some letters go under others).
'european capital' becomes 'riga' (capital of Latvia).
'r' after 'riga' is 'rigar'.
'budge'+'rigar'='BUDGERIGAR'
'may get' is the link.
(Other definitions for budgerigar that I've seen before include "flighty type" , "Flier" , "Small parrakeet" , "Bird commonly kept as a pet" , "Australian parakeet" .)