How can I make use of iterative "work in progress" versions of papers I often find myself writing the N+1st version of the same paper: we might have a short periodic internal report, a longer report for the funding agency, and  we're now writing a conference paper.
As N gets larger, this gets more and more confusing.

While the early iterations were good, they had different purposes, lengths, and level of details. 
For example, the internal "work in progress" version of a paper might have many preliminary results, but almost no introduction.
The conference version needs to have a more detailed introduction, but fewer (more focused) results.

Currently, I find that writing paper version N+1 requires starting with a rough copy/paste from version N, printing all version until N, and then looking in all previous versions for any missing information.

What are some strategies for better handing these multiple iterative versions of these papers?

Note that in contrast with this question, we don't start with a complete result (something that could be considered the "superset" paper).
Early iterations are mostly rough drafts, and sometimes middle versions do diverge because of different audience/requirements.
Not sure if I understood you correctly, but if you need a better workflow for managing the progress and evolution of your drafts, then I think the question you've linked to actually answers your question.

A combination of git and modular LaTeX can help you to:


track changes in your draft(s)
make sure new versions (and new versions of new versions) start with everything that was in the version they were started from
merge changes between versions
split your draft(s) in such a way that it is easy to include different bits and pieces in different versions
greatly simplify collaborative work
reuse parts of your projects (e.g. git project setup, LaTeX macros/environments/styles, any compilation scripts)
and many more things