What most professional and commercial landscapers do with mushrooms, is they pick them as soon as they see them appear because it helps prevent their spores from spreading to new sites.
And while it is true that many spores are wind-blown long distances, and they can easily come into your lawn and yard again from neighboring areas, if the conditions are not right, they won't grow and form mushrooms.
The Good News:
Most lawn mushrooms do not harm your lawn nor cause any damage. They are simply, and quietly, decomposing organic matter and releasing nutrients that are then available for plant growth back into the soil, and that's good to know.