What is the proper way to ask for an unpublished paper?
Just ask politely, and with the expectation that you may be reasonably refused.

Researchers vary greatly in their relationship to unpublished manuscripts. The spectrum ranges broadly, from some who are very happy to share anything they consider "not embarrassing," to others are extremely protective. Then there are many who will be just plain unresponsive for any number of reasons.  

I would thus suggest that in your email, you say something along the lines of the following:


Where you heard about the unpublished work, and why you think it will be useful to know specifically about this unpublished work and not other recently published material.
That you would be very grateful if they are willing to share, but understand that they may not wish do to so.
That you would also be happy to receive pointers to published work that can provide approximately the same benefits.


Importantly, remember that no researcher has a moral obligation to share work that is not ready for publication.  The scientific community generally holds that work should be shared in a timely fashion only once it is reasonably ready for publication.  You can argue about how to draw that line, but if a researcher really thinks their work isn't yet solid enough to disseminate or submit, then they're usually right.  In that case, you probably wouldn't really want to build off of it yet anyway, since it is likely to change in the near future.