When you push to production from staging, it takes a few minutes for all the content to be moved, the same goes for when you push from the staging environment onto production.
Depending on the size of the website, a good general rule of thumb, if you have a rather large site is to give the transfer at least 5 minutes to complete before trying to access staging or production after initiating either.
For further information on Staging Environments, please click on the link below:
help.mshini.com/knowledgebase/what-is-a-staging-environment-and-why-should-i-use-one