I would like to comfirm more information about your current environment, since we are not familiar with OVH.
How did you configure your Exchange server in OVH, what's the Exchange server version and AD version? If OVH just works as a virtual server (just like hyper-v/vmas...), we could consider your migration is a type of Migrate mailboxes from Exchange Server
And do you want to make your environment a cloud only one(all in O365) or still want to keep the Exchange server in OVH etc...? For the cloud-only environment, we consider using cutover migration to migrate your users, however it may not like you expected make the changes smoothly. Another method is creating a hybrid environment with your on-prem server. (It is supported to decomission your on-prem server if you don't need them anymore) You could use the Mail migration advisor to get step-by-step guide to help you configure the hybrid. Note to log into your Microsoft 365 or Office 365 organization when you open this link.
If your current server doesn't like the point I mentioned above, other ways supported to migrate your users and data are mentioned below in the official document as well: Use the Import Service to migrate PST files or if the current server is imap supported, use Migrate email from another IMAP-enabled email system
And for the mail redirct, in Exchange online we could use mail flow rule to meet such requirements: Mail flow rule actions in Exchange Online. There are many types of recipient/groups types in o365: Recipients in Exchange Online and Compare groups
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