From a disease standpoint, I typically see this sort of symptomology if a) there are canker fungi active in the affected branches, or b) there are root rots that inhibit water uptake leading to dieback. The oozing that the client is observing could associated with a canker disease. Either way, I'd need to see a sample to make a better diagnosis.
An insect infestation is also a possibility, but Phil Pellitteri can comment more on this.
Brian (:))