To sum it up, Facebook business pages have 24 hours to respond to an user after that user has gotten in touch with their business page. Business pages won't be able to reply that user outside of the 24 hour period, unless they’re using ad campaigns.
When an agent tries to reply such Facebook ticket directly through LiveAgent he will see this error:
Sending message failed. Can not send message: Facebook authentication error. Message: (#10) This message is sent outside of allowed window. Learn more about the new policy here: https://developers.facebook.com/docs/messenger-platform/policy-overview
Facebook is currently working on a special permission type called Human Agent tag which will extend this time to reply to 7 days, however it's currently in a closed BETA. Therefore all we can do right now is wait until it gets implemented so we can start using it.
Official and more detailed info regarding these new messaging policy can be found here: https://developers.facebook.com/docs/messenger-platform/policy/policy-overview