How will climate change affect the latitudinal distribution of precipitation over each latitudinal region?
The continued trend of a warming climate that is expected due to increased GHGs will lead to a net increase in total moisture, since the atmosphere can hold more water vapor at higher temperatures.  In general, the equatorial zones and poles will have a net increase in precipitation while lower mid-latitude zones will have reduced precip, with the rain belt moving poleward.   Climate models generally show a pole-ward shift of extra-tropical precipitation because the increased heat energy allows convective systems in low latitudes to loft the convective air higher up, which in turn leads to greater horizontal transport (generally poleward) before air completely descends and contributes to the development of an associated storm system. You can also expect the poleward shift since cold temperatures accelerate the precipitation process, and since systems will have to travel a little further poleward to reach those cold temperatures.

A 2014 IPCC Technical Summary figure (below) shows change in precipitation forecasted ~65 yrs into the future.  The image on the left represents a best-case scenario if we reduce carbon emissions while the right represents more of a business-as-usual scenario.