There is no formal process that I'm aware of, for a "complete" flush including heater core. Here's the FSM doc which isn't very helpful (and, has wrong capacity for the W124/M119):
What I do is drain the radiator and open both block drains, then disconnect a heater core inlet hose and gently blow compressed air through the heater core. This doesn't remove all the coolant but it pushes some out. Normally you do not need to "flush" the heater core, or any of the cooling system, but if that is desired you refill with plain water, run to operating temp with heater on max, allow to cool, drain, repeat as needed until you are happy. Then refill with the desired percentage antifreeze pre-mixed.
The standard dealer procedure is, AFAIK, to simply drain and refill - they do not normally perform any "flush". However, I don't know if there are fancy machines now which might be used to flush the systems? If so, I also don't know how effective they are.
@jhodg5ck ?
@Klink ?