Downfall caused by murder in a French gîte, primarily (7)
I believe the answer is:
undoing
'downfall' is the definition.
(both can mean a loss of power or position)
'murder in a french gite primarily' is the wordplay.
'murder' becomes 'do in' ('do in' can mean to kill).
'in' is an insertion indicator.
'a french' becomes 'un' ('a' in French).
'primarily' suggests taking the first letters.
The first letter of 'gite' is 'g'.
'un'+'g'='ung'
'doin' inserted within 'ung' is 'UNDOING'.
'caused by' is the link.
(Other definitions for undoing that I've seen before include "Cause of ruin" , "(A person's) ruin or downfall" , "Ruin - disaster" , "Taking apart" .)