Can anyone suggest what might be killing this cotoneaster?

Hard to say since the bit you show appears to be coming into leaf fairly
normally. Need to see the diseased plant material, but it is susceptible
to fireblight which does characteristic damage and is obvious at this time
of year with parts looking like they have been scorched by fire and then
becoming dead wood quite rapidly.

http://www.bing.com/images/search?q=...electedIndex=1

snip

That was an interesting set of pictures Martin, thank you. Thank
you also to the other posters for their helpful suggestions.

I spent some time today disentangling and removing the entire plant
and found parts that had been dead a long time that I had not noticed
before. A fungal infection seems to have been the culprit as there
were white patches on the soil also, possibly the rotting root from a
long gone broom. I decided to leave the stump for when my husband
has time to remove it, as there is a power cable (to the outside lights)
between it and the wall.