Coming from Softimage there was button for both parent and constraint compensation. When I wanted to move parent without moving it's child or editing constrained object offset, I just turned it on and made my fixes.
For constraints, official "Softimage to Maya Bridge" says that Maya has Modify Constrained Axis command for this, but when I run this after I moved my constrained object around, it just snaps back to the original offset. It seems to be the command for, like it says, modifying the axis, not the offset itself.
So in short:
1) How can I modify contraint offset?
2) Is there a way to move parent without it's children?
Thank you,