In 1871 I search for DOONER but only get DOWNER & DEGNER.
That is strange because he is transcribed as William Dooner on both Ancestry and FindMyPast. I just did a simple search for William Dooner on Ancestry and he came out top of the list

To find him I noted that the two daughters with him in 1881 had been born in Stockton. So it seemed possible that he'd been living there in 1871.
So i searched for a William D*r in Stockton born c. 1836. Dooner looked like a possible match, and so it was
In 1891 I get "Your Search for Doverner returned zero good matches".
Did you search using the census reference I gave? That should find him.
If you are using Ancestry you will find him as William Doveanes.
I found him on FindMyPast where he is transcribed as William Doverner.