I don't know of a source for MIs, but if you are looking for burials (rather than additional info from gravestones) then the Blackhill Cemetery records are online on Family Search if that's any help?
http://www.rootschat.com/links/01mp3/Years ago Durham Cemeteries used to have links to .pdfs of burial listings in various cemeteries, but they seem to have taken them down :-(.
When they were there, I downloaded the files and Blackhill was one of them. I don't think I'd be allowed to post them but I am sure I can do lookups - so I can see the following when I search the file for Bulmer :
Entry 1485 Sarah BULMER , wife of Richard, age 81, Place of death Leadgate, buried 02 July 1870
Entry 3259 Richard BULMER, gatekeeper, age 85, Place of death Thirsk (previously of Delves Lane), buried 10 Jul 1877
Entry 3698 Ann BULMER, wife of Joseph, age 33, Place of death Consett Terrace, buried 08 Aug 1879
Entry 6161 Sarah Ann BULMER, spinster, age 65 ,Place of death 11 Front St Consett, buried 14 Sept 1889
Entry 24803 Amos Bulmer STOCKTON, age 98, Place of death Villa Real Consett, buried 24 April 1952
Those should help you find the relevant images on Family Search.
If this is of use, then I'll look for more Stocktons tomorrow if you like
Boo