Each mine is loaded with 2,000 stone/gold, so they do run out rather fast. However they replenish with resources at I think +200 per day. We might speed this up in the future.
Dawn of Fantasy`s economy was meant to be fluid - so you don`t just place some peasants on one task and they`ll keep on doing the same thing for the next 2 months. You can see this approach with mines running out or with farm fields unavailable during winter