You are not going to damage the transmission. There are only two things you could have issues with. The differential on an AWD vehicle, and the traction control system on all newer vehicles. In general, if you can keep the rolling diameter within 1-2%, it should be OK (I don't recommend it, but if you do it, stick to this range). The generally accepted spec is typically 3%, but that is too much on modern cars with advanced traction control systems and differentials.
Most traction control systems perform some function at around 5% of detected wheel slip, and starting with over half that at 3% is too much. As far as the differential, same thing, but more of a mechanical concept. The differential(s) are looking for wheel slip. More detected slip (i.e. different rolling diameters) will cause more wear on the differential as it is constantly trying to compensate for that difference in rolling diameter (seen as wheel slip). Obviously the closer in size the F/R wheels/tires are, the less wear you will see. Again, keeping the tires in that 1-2% is not going to cause immediate problems, but possibly long-term ones if that is a concern for you.
There are two other dynamics that are not covered by the above. There are several high performance AWD vehicles that do come with staggered wheels/tires, and in those cases, the traction control is tuned for those specific differences, as is the locking action/power distribution of the differential. That is obviously not the case for a vehicle that comes from the factory with non-staggered wheels/tires. The other part is handling dynamics. This is maybe not all that important on a non-performance vehicle like the Explorer, but wider rear wheels generally induce more understeer. From a performance standpoint, most people would consider this undesirable. Also, stating the obvious here, you lose the ability to rotate your tires....
The moral of the story is that you can do it if you follow a few key rules in terms of rolling diameter, but there are some possible adverse effects whether one wants to acknowledge them or not. It's definitely not going to ruin your car in 6 months if done right, but like most things you do to your car, you should be aware of all the factors related to such changes.
Edit: The 265/40/22 and 285/40/22 setup on 22x9"/10.5" is just over 2%, and probably close to the acceptable range (again, I would normally recommend under 2%, but it's close). The difference on that setup is 2.1% on the tires alone, and maybe a little lower with the extra few mm's of stretch on the rear tires (20mm's of tire stretched over an extra 38mm's of wheel)....
