Lag first to find completely live deer (4,6)
I believe the answer is:
fall behind
'lag' is the definition.
('fall behind' can be a synonym of 'lag')
'first to find completely live deer' is the wordplay.
'first to' says to take the initial letters.
'completely' becomes 'all' ('all' can be a synonym of 'completely').
'live' becomes 'be' ('be' can be a synonym of 'live').
'deer' becomes 'hind' (hind is a kind of deer**).
The initial letter of 'find' is 'f'.
'f'+'all'+'be'+'hind'='FALL BEHIND'
(Another definition for fall behind that I've seen is " Lag".)