Everyone, say, runs over, initially moving quickly (7)
I believe the answer is:
allegro
'moving quickly' is the definition.
(musical term)
'everyone say runs over initially' is the wordplay.
'everyone' becomes 'all' (all people).
'say' becomes 'eg' (both can mean for example).
'initially' suggests taking the first letters.
The first letters of 'runs over' is 'ro'.
'all'+'eg'+'ro'='ALLEGRO'
(Other definitions for allegro that I've seen before include "A fast piece of work" , "Fast (mus.)" , "Quick (in music)" , "Austin" , "Brisk and lively tempo, musically" .)