Recently Microsoft announced that mailbox delegation would be available between Cloud and On-Premises accounts. This would allow for a cloud mailbox user to send-as an on-premises mailbox.
Looking at the documentation (https://technet.microsoft.com/en-us/library/jj200581(v=exchg.150).aspx) it appears that this should now be working in early May 2018. In particular.
|As of February 2018 the feature to support Full Access, Send on Behalf and folder rights cross forest is being rolled out and expected to be complete by April 2018.|
This feature requires the latest Exchange 2010 RU, Exchange 2013 CU10, Exchange 2016 or above but otherwise should just work.
Unfortunately when a user tried to use this feature it didn’t work.
This message could not be sent. Try sending the message again later, or contact your network administrator. You do not have the permission to send the message on behalf of the specified user. Error is [0x80070005-0x0004dc-0x000524].
Notice that this mentions the send on behalf rights. Well in this case the user didn’t have those but instead had the more powerful Send-As rights.
Well it looks like Microsoft are running a bit late on the rollout with this other article (https://technet.microsoft.com/en-us/library/jj906433%28v=exchg.150%29.aspx?f=255&MSPPError=-2147217396) now shifting the rollout completion to Q2 2018.
|As of February 2018 the feature to support Full Access and Send on Behalf Of is being rolled out and expected to be complete by the second quarter of 2018.|
Either way it’s not much longer but in the interim you may need to keep assigning send on behalf rights prior to migrating mailboxes. This will save you having to use powershell to do this post-migration since the on-premises ECP interface doesn’t support granting these rights to cloud mailboxes.