Answer: Broom (Cytisus) is somewhat likely to die back the way you described, and pruning out the dead material is the right response. For this reason, it is difficult to keep them looking symmetrical. Routine care would include trimming them back hard after bloom each spring. This helps a little bit with size and tidiness but older plants tend to be scraggly and occasionally die out unpredictably.
I am not sure if you would be able to transplant them successfully since they have very extensive root systems, their reputation is of being difficult to move. Your better bet might be to enjoy them this spring and then remove them, try the transplant if you wish, but be prepared to replant with new plants next spring if necessary. You might also consider another location for them if they are overgrowing the allotted space.
I'm sorry I can't be more encouraging, I love this plant in bloom, too.
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