Dope runs across grass becoming increasingly active (6)
I believe the answer is:
spryer
'increasingly active' is the definition.
'dope runs across grass' is the wordplay.
'dope' becomes 'sp' (betting slang for SP, starting price).
'runs' becomes 'r' (cricket abbreviation).
'across' indicates putting letters inside.
'grass' becomes 'rye' (I've seen this before**).
'sp'+'r'='spr'
'spr' enclosing 'rye' is 'SPRYER'.
'becoming' acts as a link.
(Other definitions for spryer that I've seen before include "More lively" , "more agile" .)