Escapes with a bachelor’s degree on, initially, Dairy Science (8)
I believe the answer is:
absconds
'escapes' is the definition.
'a bachelor's degree on initially dairy science' is the wordplay.
'bachelor's' becomes 'B's' (abbreviation e.g. BA=Bachelor of Arts).
'degree' becomes 'c' (C is an example).
'initially' indicates taking the first letters.
The initial letters of 'dairy science' is 'ds'.
'a'+'bs'+'c'+'on'+'ds'='ABSCONDS'
'with' is the link.
(Other definitions for absconds that I've seen before include "Bolts, runs away" , "Decamps" , "Does a bunk" , "Flees, runs away" , "Flees to escape from a legal process" .)