The design of the Moulton is very old and does not take into account pedal induced BOB. The Birdy does. As the chain pull forward on the rear hub the Moulton will move. The Birdy is designed to have the rear suspension pivot inline with the chain. Nearly no pedal induced BOB. However your pedaling motion while standing will cause both to bob...
The mountain bike world discovered this a long time ago and has moved the rear suspension pivot (on single pivots anyway) inline with the chain.
I have never ridden either of these bikes but I have had over a dozen different full suspension mountain bikes.