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Items such as design, store settings, brands, categories, etc., are moved within seconds but larger items like orders can take consume significantly longer time. Anonymising data will further slow down the process due to the extra steps involved.
While the whole point of a staging environment is to match as close as possible to the production version, there are cases where copying all your orders simply to make a design change on your homepage is a complete overkill. Think about what you are moving and why, before creating a new environment.