Rushed to swallow last of aioli dip, showing precipitation (8)
I believe the answer is:
rainfall
'precipitation' is the definition.
(I know that rainfall is a type of precipitation)
'rushed to swallow last of aioli dip' is the wordplay.
'rushed' becomes 'ran'.
'to swallow' means one lot of letters goes inside another.
'last of' suggests the final letters.
'dip' becomes 'fall'.
The last letter of 'aioli' is 'i'.
'ran' enclosing 'i' is 'rain'.
'rain'+'fall'='RAINFALL'
'showing' is the link.
(Other definitions for rainfall that I've seen before include "Climate measure" , "Shower" , "Amount of water coming down" , "Water falling in drops" , "Precipitation, get out umbrella" .)