377s & all sub-classes can all work with each other. Technically the whole 377/387 fleet are compatible with one another, but I believe 377s & 387s together can't work in passenger service due to software incompatibilities.
377/6s and 377/7s subclasses are near identical to 387s, excepting compatibility with the older 377s. Bombardier rebuilt their entire TMS software stack (formerly windows xp, then Linux), and I am told that the interoperability was no mean feat.
I would speculate that the older PISs are tightly coupled to their destinations and routes, making multiple working with other classes infeasible in passenger service, but newer PISs may be more configurable.
I am particularly curious as to whether class 379s and 387s are compatible.