Game birds finally heading for one of several rivers (6)
I believe the answer is:
grouse
'game birds' is the definition.
(grouse is a type of game bird)
'finally heading for one of several rivers' is the wordplay.
'finally' says to take the final letters.
'one of several rivers' becomes 'Ouse' (several rivers are named the Ouse).
The final letters of 'heading for' are 'gr'.
'gr'+'ouse'='GROUSE'
(Other definitions for grouse that I've seen before include "See game bird" , "Plump game bird" , "Grumble like game-bird" , "Complain pettily" , "Grumble; game bird" .)