In the 2.0.3173 update, the corresponding What's New post mentions that [fully-constrained] sketches can now be copied and pasted between documents. This is a great usability improvement.
However, it brings up an issue that I think users will begin to encounter: The Copy command copies the Parameter Name if it is used in a dimension in the source design. If that same Parameter Name exists in the target design, the name will be preserved when pasting, otherwise the value of the parameter will be used. This can cause very undesired behavior if the Parameter Name exists in the target design with a different value. I know that in my own workflow, I have a common set of parameter names that I tend to re-use for various aspects of parts in different designs (ex. post_length, case_height, etc.), and I don't think I'm alone here—and that's where the problem arises. To be clear, I think that copying parameter data is the right way to do it, but the problem is that the parameters themselves are not copied in the process.
The community has been asking about proper support for shared parameters for a long time (ref, another ref). I'm talking about shared parameters like in Revit. I know there is the Parameter I/O plugin for Fusion 360, but that is just a time-consuming workaround and not the kind of support I'm describing. The introduction, in this update, of the ability to copy and paste parameter names as sketch data makes a much stronger case to provide shared parameter support in Fusion 360 now.
I hope someone on the product team is listening!