I'm not sure what you precisely ask about. Can you please clarify.
It's a 21 minute video, you have posted a link for.
About LFO phase.
In Pigments you can modulate the LFO's Phase parameter. In example using the KBD and Velo modulators. That together with the Poly KBD you mention will set different phases for each voice, if you play a full chord instead of playing a single voice in the chord at a time.
Playing the voices randomly after eachother would also give different phases for the LFOs per voice, but i assume you know that allready.
Using in example a Random Module with Sample and Hold set to be triggered by Poly KBD may also offer some possibilities, depending on your needs.
It could in example be nice if Pigments had a VOICE NUMBER modulator.
You can do many things with Pigments LFOs. Also try to use the KeyTrack option a LFO have to modulate it's rate.
Also keep in mind, that you can nearly modulate anything with anything and so on. Sometimes an effect done in one way in one synth, can be done in another way in another synth, if they can't do it the same way. Can't tell if this is the case for the effect you wan't.