Family brings up quote that's moving (7)
I believe the answer is:
kinetic
'moving' is the definition.
(similar in meaning)
'family brings up quote' is the wordplay.
'family' becomes 'kin' (synonyms).
'brings up' shows that the letters should be reversed in order (I've seen 'brought up' mean this (in a down clue, letters go up)).
'quote' becomes 'cite' (synonyms).
'cite' reversed gives 'etic'.
'kin'+'etic'='KINETIC'
'that's' acts as a link.
(Other definitions for kinetic that I've seen before include "Energising" , "(E.g. energy) of motion" , "Pertaining to motion" , "Resulting from motion" , "Caused by motion" .)