One from Central America rushin' up, pursued by an exotic bird (10)
I believe the answer is:
nicaraguan
'one from central america' is the definition.
(someone from Nicaragua)
'rushin' up pursued by an exotic bird' is the wordplay.
'rushin'' becomes 'racin'' ('racing' pronounced like 'rushing' was; 'race' can be a synonym of 'rush').
'up' is a reversal indicator (in down clue: letters go upwards).
'pursued by' becomes 'a' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'an exotic bird' becomes 'guan' (bird of the Americas).
'racin' back-to-front is 'nicar'.
'nicar'+'a'+'guan'='NICARAGUAN'
(Other definitions for nicaraguan that I've seen before include "Central American" , "Latin" .)