This is probably a little backwards (in that I should have read Prince's direct brief first), but I had a chance to look at the amicus brief filed yesterday on behalf of the Warhol Foundation in support of Prince's position. The Foundation is being represented by the Stanford Fair Use Project, along with Virginia Rutledge and others at Bingham McCutchen in New York. You can read the brief here.
The main argument is that the district court got the transformativeness analysis wrong in two ways. First, it said that Prince's work had to comment specifically on Cariou's photographs; instead, "all that is required is a meaning, message or purpose that is 'separate and distinct' from the original." Second, it "assessed the meaning of Prince's work based entirely on Prince's testimony, not the reasonable perceptions of the viewer"; transformativeness should be determined "first and foremost by the observation of the work itself, and whether new meaning and expression may be reasonably perceived from it."