It’s really hard to make out what’s going on there but it appears that one of the grafts is failing. I’d prune that section off.

floral-

What are the characteristics of a failing graft in a tree that's more than 10 years old?

Sorry the pictures aren't more descriptive, but it was the best I could do. It looks like a normal, grafted plum tree on the other branches, some of which are in the right of the first photo; the second photo is just a closeup of the area with the dense leafing and the slightly weird branching. It's just that one major branch has all this dense growth filling in. I have a much larger plum tree, a Stanley plum, that is about 25' high and is healthy and bears a lot of fruit. It has one minor branch in the lower area of the tree where this type of weird growth is also occurring, but 99% of that tree looks normal.

Dying branches issuing from one point would indicate a failing graft.

Not all varieties of plums have the same vigor and dwarfing root stock will only grow a small tree that is supposed to produce fruit sooner, thin out the dense graft, keep what is growing in the direction that you want. it may produce water sprouts if you remove a lot but there is nothing to lose at this point.

floral- The area that got my attention is extremely rigorous, which makes me question that it’s a failing graft. I will, however, see what the graft looks like later this afternoon, and post back.

Here are some more photos:

The first one is a closeup of the dense foliage. The second one shows the underside of that leaf mass with the tangle of small branches. The third photo is the branch (going up and to the left) that all this growth is on, and it doesn't seem to show any signs of failure at the graft junction.

Further ideas welcome....