Pause in relationships and constraints I want to talk about relationships between (for example) two bones. There are two bones (A and B), bone B has a Cconstraint, bone A is the target of that Constraint.

Let the Constraint be "Child Of". When we reduce the influence of the Constraint to 0, B folllows no more transforms of A (and that is what I want), BUT when we increase the influence back to 1 B "jumps" to A's transforms. So it collects all the information about A's transforms, and at zero point we just don't see it, but it exists and has no interrupts.
Is there any way to stop any inherriting of A's transforms for B, so that we move A, and B also moves (at 1), then B stops (at 0), and then B again moves (at 1), BUT from point where it has stopped last time? With no jumps to A's actual transforms.

It is about all the constraints, no matter is it Child Of, Copy Transforms etc.

Any solution (scripting, parenting, other consraints) would be great.

P.S. Drivers also have the same principle so I'm stuck at this point.
It doesn't collect all the transforms, it just uses the relative offset, that the target had have at the moment you created the constraint. For the Child of there are two buttons 'Set inverse', and 'Clear inverse'. They change that stored offset. So to get what you ask, you need to change the offset. I have no idea, where and how it is stored in the constraint data, but I'm sure it's accessible from the python API.