Past the point of comfort
There's a point in many techniques (heck, maybe all of them if I thought about it more, but I'm thinking of some specific recent examples here) when uke reaches the end of his range of motion. We talked somewhat about this with kote gaeshi, when a handful of people were struggling to make it "work".
They had the hand position, they kept their centers moving as we're so often told, they maintained ma'ai, all of that. Uke was slightly off balance, or at least his posture was a little bent, but didn't fall. He just continued to stumble along.
If that sounds familiar, try taking your partner and just stand there, not moving around, but facing each other. Take his hand in a kote gaeshi grip, and without moving around, just arms, go from the starting point moving your arms in an arc until you get to the "end", the kote gaeshi. If you're tori, you should be able to feel all of the slack is taken out of uke's arm, everything's tight, and his arm just stops moving. He's moved his arm to the very edge of his range of motion. And even though his wrist is somewhat contorted, he can tell you he's actually still fairly comfortable.
I actually see that a lot with many techniques, and in fact, is exactly what I did for the loooooongest time. Oshi taoshi, ude gaeshi, and shiho nage are good examples, too. I would get uke to the point where all the slack was taken out, where I felt everything tighten up, where uke wants to stop, and I stopped too. Posture slightly broken, but no fall, no throw.
Well, I thought, I'm always told to keep my center moving at all times, don't stop, so maybe that will finish it for me.
Unfortunately, uke never fell (unless he was just being compliant) but just continued to stumble after me indefinitely. Finally, I realized I'm just making things convenient and comfortable for uke by stopping when his arm is ready to stop.
Go back to that little exercise I was describing earlier: standing still, moving uke's hand back and forth in that arc, from beginning to end, to the point where you feel you've taken out all the slack and uke's arm just naturally stops. Do that a couple of time, and then at some point, take uke's hand just an inch or two beyond that point.
Even standing, I think you'll find uke's posture goes into catastrophic failure and he drops pretty readily. I didn't force it, I didn't crank on it—you don't need to. Whatever uke does, you don't resist, you follow, and keep following, and when he stops, you keep going and take him a little further than where he expected to stop, right? Sounds pretty obvious, but I still missed it and I still see it happen still. Do the same stationary drill with oshi taoshi, ude gaeshi, and shiho nage.
So where does the bit about keeping a constantly moving center come in, if you can clearly create the conditions necessary for a throw while standing still? The main reason I can think of is that if I stop my ass, and uke continues to move his, and uke knows what he's doing, he can very readily reverse it and dump me on my head. I suggest doing this while standing still as merely a drill, to isolate the one thing we want to focus on, so students don't have to try and think about their feet, etc., too.
Stopping, I believe, will also "lead to the dark side" so to speak, and the temptation to use strength and crank on a wrist, or ikioi (arms moving independently of the center, and many times implies the use of muscle strength) which has a spotty success rate (little ladies trying to do it to giants, or guys who just don't respond to pain and believe me, they're out there) When we take uke's wrist (or whatever) past the point of uke's range of comfort, we still need to have our moving centers behind it.
Like I always say, this probably isn't news to many of you, but it was a critical realization to me!