When an email client is set up using the POP3 mail protocol, it downloads the latest email it sees when it hits the mail server, and the server then deletes it. When the client is set up with the IMAP4 mail protocol, it downloads the email but the server retains it, where it can be downloaded by the next client set up for that protocol. That'd be something to check in your Outlook clients, because if some of them are set up with POP they'll be the only ones to get any specific email, where the ones set up with IMAP will download and it'll still be on the server when the next client logs in, which sounds like your symptoms.
Best of luck!
All the emails say POP3
With the POP3 protocol, it is possible to tell the server to retain the email.
Whether the version of Outlook the OP is using offers this option is another question. But in a POP3 protocol dialogue, the download of the email and the delete of the email from the server are two separate commands.