Starting an on-prem to Cloud migration with a hybrid configuration. On-prem users can e-mail anyone just fine. Cloud users can e-mail externally and other cloud users, but not on-prem users. E-mails from cloud to on-prem users do not error, but rather appear to be sticking in the cloud and not be synched with on-prem. If an on-prem user logs into the cloud user interface via the tenant link, the e-mails from the cloud users are there (but they can't see the on-prem e-mail in that view).
I've looked at mail flow and user SMTP settings in general as it seems like something is disconnected, but it's kind of blind poking around. Internet searching only finds people where the mail actually errors out. In my case there's no error, it delivers, just not to the right place!
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Additional information: Trying to test the O365 to on-prem connector with on-prem users fails. It reports that the on-prem user is in O365, so can't send. However, if I inspect the mailbox it is clearly marked as on-prem.
Best Answer
According to your description, if the user login Office 365 OWA, he can see the email, we could know that the mail flow is working fine. And now this issue we need to confirm is the type of mail flow, check if it is from cloud to on-premise server. How did you check the mailbox? The accounts can be synced to cloud but the mailbox cannot, you can migrate it to the cloud. In other words, there is only one mailbox, there is no sync. You can run get-user in EMS of the on-premise server, please check if it is Mailuser or UserMailbox. If it is mailuser, it means that its mailbox is not on on-prem.