If your puppets are structured EXACTLY the same way, you could do all of your warp independent(+) choices and keyboard triggers in AI or in the Puppet panel. Then when you're done, you could duplicate that puppet in the Project Panel, then re-link it to different AI footage. You can do this in the Properties Panel with the footage selected in the Project Panel.
This will only work if the puppets are structured the exact same way. Once you've linked your different AI files, you could then rig them differently with Sticks and Draggers.
All that said, you're probably best off just rigging them totally separately which is annoying, but far less likely to introduce weird errors.
Dan R.
CH QA