Potential workaround for two issues with Master Properties
Specifically, the inability to copy keyframes or set a nested comp's Master Properties as a MP in a parent comp. Create a solid/null/something invisible, and apply multiple Effect Controls corresponding to each MP (Slider, Angle, Checkbox, etc.), then link the MP's to them with the new pickwhip.
It's a pretty clunky workaround, and not robust at all, but it's better than nothing (arguably).