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Hi Terry,
Based on your situation, the workable scenario is hybrid deployments. Some articles about deploy hybrid and move mailboxes between on-premises and Exchange Online organization in hybrid deployments below for your reference:
- Hybrid Deployments
https://technet.microsoft.com/en-us/library/gg577584(v=exchg.141).aspx
- Hybrid Deployments with the Hybrid Configuration Wizard
https://technet.microsoft.com/en-us/library/hh529920(v=exchg.141).aspx
- Move mailboxes between on-premises and Exchange Online organization in hybrid deployments
https://technet.microsoft.com/en-us/library/o365e_hrcmoverequest_fl312271(v=exchg.150).aspx
Feel free to let us know if you need further assistance during the procedure.
Regards,
Larry