Dog starting to tire and stray? That's right! (7)
I believe the answer is:
terrier
'dog' is the definition.
(terrier is a kind of dog)
'starting to tire and stray? that's right' is the wordplay.
'starting to' suggests taking the first letters.
'and' means one lot of letters go next to another.
'stray?' becomes 'err' (to err is to stray or go wrong).
'that's' becomes 'ie' (abbreviation for 'id est', 'that is' in Latin).
'right' becomes 'r' (common abbreviation).
The initial letter of 'tire' is 't'.
't'+'err'+'ie'+'r'='TERRIER'
(Other definitions for terrier that I've seen before include "See 11" , "Dog originally used to ferret out burrowing animals" , "Breed of dog - volunteer soldier - estate register" , "Reserve soldier" , "(Yorkshire?) dog" .)