Hi @Kris Callahan ,
Rules where are they stored in hybrid mode and how to get them to 365 cloud when we remove on premise
The Outlook Inbox rules for an Exchange on-premise account are stored in the mailbox and can be migrated together with the mailbox itself to 365 cloud. Therefore normally you don't need to worry about the Outlook rules when migrating to Exchange Online from Exchange on-prem.
Here's a thread which discussed on the storage location of inbox rules for your reference:
Rules stored on exchange server, even though they're client-only rules
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