Mosses are encouraged by low fertility, acid soils, and compacted soil. These are things that most plants generally dislike, and so the moss has less competition, and won't get overrun by taller plants.Soils can be be dry or wet, shaded or in sun, which can determine which mosses will thrive. Mosses don't have roots like regular plants, so soil depth isn't important, as long as the moisture retention that the species needs is there.
If you think you have found a suitable place, you can take 4-6 inch (minimum) diameter patches and transplant them. Take a layer like a pancake or biscuit that includes a quarter inch of soil. More soil is fine, but it is not needed. Transplant to the new location, placing it at the same level as before. Do not put soil on top! Water with water that is neutral or acid (pH 7 or less). Do not water with city water. If you have a well, that should be fine, or you can use distilled water, creek or lake water, or rain water. This first water should be all it needs. If it likes it there, it will thrive. If it dies, then mostly likely any further attempts will fail also.