So, are wood shavings a good starting point on an earth floor in a (can be wet) run? I saw some posts say they don't breakdown well?
What should the next layers be and do you wait a while to add each layer?
In the run, unless it's covered and small, it's not as important what you use and when, though it's still a good idea to use a variety of materials that have different break down times. Wood shavings take a very, very long time to break down, especially in a coop setting, but they can be a good base for a run if it has a lot of soft mud going on. If the run is covered and very small I'd treat it much like a coop and use materials that break down more quickly, along with those that provide air spaces.
In a run that is uncovered you don't need to layer it in as slowly as you do a coop....you can make it as deep as you wish in a run, as long as you have high sides to hold in the litter. You'll still want it to compost there, so using a variety of materials works best, but you can go deeper, faster to help keep the chickens out of the mud. The deeper the litter in the run, the better those layers next to the soil will compost, attract bugs and worms and wick moisture away from the top layers. If you use all wood shavings there, it will take a very long, long time to decompose and meanwhile it may put off ammonia smells during the hot and wet parts of the year.
Leaves are great, small twigs, pine cones and needles, bark, corn shucks and other garden debris/trimmings, grass clippings (not too many for the space if you live where it's mostly humid....nice to let those dry and then put them in the run), a little of this and a little of that seems to work best.