In fact, the Church has already gone specifically moderate on quite a few issues; in Utah they intervened to show their support for the moderate immigration proposal called the Utah Compact, and in Arizona, there's rumors that they quietly supported anti-immigrant Mormon Republican Russell Pearce's recall opponent (also a Mormon). They also supported an anti-discrimination law in Salt Lake City. Additionally, Orrin Hatch's political survival may be in part due to the LDS Church's specific admonition over the pulpit that Mormons should go out to vote (which also repeatedly pointed out that "there is truth in all political parties and the Church does not endorse any of them"). More moderate delegates turned out at the Utah GOP convention in 2012 than in 2010. Conversely, the LDS Dems went from being formed in mid-2011 to having over 2,000 members in 2012.
Heck, though they haven't said anything, the LDS Church's move to have more energy-efficient and environmentally-friendly buildings is an obvious move towards moderate environmentalism.
The LDS Church will become more politically moderate, just give it time.
That conclusion is contingent on the assumption that our definitions of "liberal", "moderate", and "conservative" do not change. Which may not necessarily be true. After all, I love the Utah Compact a lot, I dislike SB 1070, and I like Orrin Hatch, but half of this board probably considers me a fascist.
And plus, there's already plenty of Mormon Democrats. Harry Reid, for example. I don't think it's a new phenomenon.
