Go to the dogs locally last month with family (7)
I believe the answer is:
decline
'go to the dogs' is the definition.
(thesaurus)
'month with family' is the wordplay.
'month' becomes 'dec' (abbreviation for December).
'with' means one lot of letters go next to another.
'family' becomes 'line' (I've seen this before).
'dec'+'line'='DECLINE'
'locally last' is the link.
This may not be right. Some or all of it may be part of another bit of the clue.
Can you help me to learn more?
(Other definitions for decline that I've seen before include "Fade" , "Refuse - to take a drop" , "Downturn - turn down" , "Become fewer; politely refuse" , "Refuse - dip" .)