The concern isn't the transmission like many 'less aware' mechanics think. The concern is the increased rate of wear on the diff/s. If you have clutch based LSDs then it will use the clutches up at a very fast rate. If you have a Torsen it won't use it quite as fast, but they to are capable of wearing out (mine has, though from use not different tires).
The gears and/or clutches inside a diff are not designed for constant velocity, but low speed sporadic movement in both directions. With two different sized tires you are looking at constantly going one direction 100% of the time.
All in all though, 99% of all cars/SUVs are able to deal with these small discrepancies without even noticing. But there are a few vehicles out there with lockers and heavy duty diffs that do not take well to this situation, and have spread their story like a wildfire in the plains of Utah.