Graduate, extremely grasping, with time to get suitcases (7)
I believe the answer is:
baggage
'suitcases' is the definition.
(I've seen this in another clue)
'graduate extremely grasping with time' is the wordplay.
'graduate' becomes 'BA' (someone with a Bachelor of Arts degree).
'extremely' says to hollow out the word (remove centre letters) (only the extremes of the word are used).
'with' means one lot of letters go next to another.
'time' becomes 'age' (an age is a period of time).
'grasping' with its centre taken out is 'gg'.
'ba'+'gg'+'age'='BAGGAGE'
'to get' is the link.
(Other definitions for baggage that I've seen before include "Travellers' luggage" , "Suitcases, travel gear, trunks etc" , "belongings" , "Naughty lady" , "reclaimed in Carousel" .)