I've never cleaned a metal working machine like that, but the channels and (and plumbing if there is any) which is different, but not that different in their design or operation than the bearings, bushings, trunions, and such that I care for pretty regularly. My opinion is that you can ALWAYS "flush" them to take new lubricant, to some degree, and in the case of a rotating bushing, that may well be enough, but you can NEVER flush them completely. So when I translate that in to longer, straighter channels, I don't think the motion of a table, knee, or any linear slide really is going to get "flushed". Part of it will for sure. You'll see "flush" come out, and it'll bring some grease with it, and then you'll see it clean up. But I don't think you can "force" all the leftover binders out of there to a degree that will get the oil to everywhere it needs to be.
So my opinion (as a mechanic/technician very familiar with having to flush grease from things), is that this will be more likely to lead to a false sense of success, and probably a partial success on any given sliding surface, but I highly doubt you'll get it all. And of course you won't have any way to know what percentage you did or didn't get, where the oil is in the spots you can't see, where it's still blocked, so you'll have very limited ways to guestimate as to what's going to be adequate.... I'd take it apart.
