Migrating On-Prem Distribution Groups to Exchange Online in Hybrid Mode

You should recreate the DLs in the cloud and then remove the on-premises equivalent. People often script this removal so that the new cloud DL receives the old email address used for the on-prem DL.  You could do it like this:

Create the new DL in the cloud. Set the new DL to be hidden from address lists so no one can see it. Give it a temporary email address.

Add the members to the cloud DL.

On-premises, update the old DL to give it a new email address and hide it from address lists.

After the on-premises change synchronizes to the cloud, update the new DL to give it the original email address and unhide it from address lists. Clients will now see the new DL and begin using it. 

After a week or so, when you are sure that everything works as expected, you can delete the old DL from on-premises AD.

Email addresses always need to be resolvable before delivery can occur. Once you switch the MX records, the cloud DLs can be used.