Well I just went for a quick ride on a workmate's dually. I tied both almost lockout and fully active rear suspension setting, and there was very little bobbing going on even at crazy cadences.
I can't do that on my Mini.
I would conclude that damping is everything. maybe geometry has some influence as well. And maybe the seatpost flexing is another culprit.
* goes away, thinking about how to increase Downtube Mini's damping... *
The geometry of the rear suspension is huge. The higher the pivot point (real or virtual) of the suspension, the more 'anti-squat' (due to pedaling) the suspension will have in it. Of course, as it get higher and higher, the path of the wheel gets less and less vertical which causes other issues.
Damping can help significantly, but it has it's downside. Damping takes mechanical energy (i.e. your spinning legs) and converts it to heat (instead of pushing you forward). I'd rather take that mechanical energy from my legs and turn it into kinetic energy of me and the bike going forward.
A lot of the problem comes from the spring selection. When pedaling at a good cadence (90 rpm, 180 leg strokes), you'll be supplying a 3 Hz input. To have a suspension with such a short amount of travel, Yan has had to stuff a pretty stiff spring in the suspension (or risk a bunch of bottoming out). Well, I'm going to bet the natural frequency of the system is around 3 Hz. So a normal pedaling cadence excites the rear suspension and you bounce up and down.
The Thudbuster has more travel and so it can use a softer spring (elastomer). This means the natural frequency might be more like 1.5-2 Hz. Well, when you're pedaling at a 45-60 rpm cadence, you generally aren't pedaling hard. You don't put much energy in the system and it doesn't get too excited (bouncy).
It would be interesting to see if a softer elastomer eliminates a lot of the pedal bob you get on the Mini. My guess is that it would, but that over small bumps you'd now have to deal with the suspension bottoming out harshly. Out of the saddle, most full suspension bikes bob a lot. Big inputs and low cadences really move things around.