Deliver and fall asleep (4,3)
I believe the answer is:
drop off
I believe this is a double definition.
'deliver' is the first definition.
(I know that drop off is a more specific form of the action deliver)
'fall asleep' is the second definition.
(synonyms)
'and' is the link.
(Other definitions for drop off that I've seen before include "Fall asleep" , "Fall < or crash" , "delivery of goods" , "Decline, decrease" .)