Hi Steven,
I think some of the points are not clear, do you have your own tenant after the split? Where are the mailboxes hosted is it onprem or hybrid setup currently? Do you own the new domain name for the Azure setup? There are lot of questions and information required for this kind of scenario .
I would suggest you follow this guidelines for Tenant to Tenant migrations - microsoft-365-tenant-to-tenant-migrations
For Onpremise migration to another domain please follow the guidelines here - cc974332(v=ws.10)