Unlike in the case of the Outlook add-in app, saving emails using mailbox sync is not immediate. Mailbox sync is a server-side integration that continuously processes a cyclic queue, which means that emails will be processed when the queue item for the user/folder is reached. Hence the time to save in the case of sync can vary, depending on how soon before your queue item is reached, you have made the move/copy. Depending on the queue and volume of users, this can take up to a couple of hours before the user/folder is processed.
Users are required to connect their mailbox before they can save emails to the Soup Mail Outlook app, click on link to get a step by step guide on how to connect your mailbox to Soup Mail.
A system administrator can also setup impersonation sync which automatically connects user's mailbox to Soup Mail.
Click here for more on Saving emails to Soup Mail.