How long has your carb apple been in the ground? If it is more than a few years you are unlikely to be able to move.

More than 2-3 years and you shouldn't attempt to move it. You can remove any of the neighbours plum tree that's overhanging your garden, but I realise you might mean its just casting shade onto your tree rather than growing above it.

I've had the crab apple for about 10 years and it has yielded a great crop for the last 5 years. This year it has been dwarfed and although the corner is south facing so it is not lacking light, the neighbours tree grows above it.

Anything that hangs over your fence line you can remove - but legally, the bits you chop off belong to your neighbour.I'm guessing they won't want them though...

Don't chop off anything without permission from your neighbor. Be sensible about this. They most likely won't mind you trimming whatever is hanging over on your side, but if you presumptuously start hacking away at their plum tree, you may become a target of revenge. Be a good neighbor.

My neighbour and I get on really well and I wouldn't dream of pruning anything without permission. I was hoping to avoid cutting anything and hoped feeding or moving would provide an answer.

Sorry Bikeydude, feeding won't help and you simply can't move... Chat to your neighbour and explain the problem and ask for permission to prune back the plum branches on your side of the boundary, if you get on well with each other shouldn't be a problem.