tips for killing a neighbours tree off slowly please.

how slow do you want? find honey fungus, drill core out of tree, insert spores. call in council as tree would eventually be dangerous and could fall.

worldclass accident is spot on with the ring barking suggestion, and to make sure it gets the message, at every point where you remove bark use a paintbrush to apply a little glyphosate (roundup)to the fresh cut area.
Alternately you could manage the tree either as a pollard or coppice it. However if as you say this is a mountain ash ie. a rowan then you must be aware of the bad luck you will bring upon yourself by harming it.

Find something worth worring about and move on 8)