I don’t know if you’ve noticed, but since Microsoft ‘upgraded’ the move to / copy to experience in SharePoint Online a few months ago, the functionality has basically been broken. No amount of calls logged to them is resolving this. The result is people having to download endless content and uploading it again in its new home. This loses all the tracking when using the Move To option.
The problem is that when you click either Move To / Copy To on the ribbon in any library :

The Quick Access list and/or More Places list either doesn’t display any sites or doesn’t display the full list. There is also no way to browse to additional site collections – a huge oversight Microsoft!


THERE IS A WORK-AROUND
Go to the source library where you want to move or copy the content from and click Add Shortcut to OneDrive on the ribbon. Then do the same on the destination library. You will be notified that it was successfully added.


You may get an error message like this, but when testing this, it doesn’t seem to be real. The shortcuts are added anyway.

Go to OneDrive – refresh your screen if you did get the above error message, and you will see the 2 folders you added. They will have large links on the corners of them.

Click on the source folder and select the documents you want to move or copy.

Click on My Files in the move / copy window.

Click on the destination library and move / copy here.

Done.

The usual considerations will apply – if the metadata or versioning settings are not the same, it will give you an error message and the option to move / copy anyway.


But that notwithstanding, at least we can get back to managing our ever-changing work environments now.
When you’re done moving your content around, you can remove the shortcuts from OneDrive.

SECOND WORK-AROUND
Thanks Rus from LinkedIn for pointing this one out. You can also pin a library to Quick Access.

Once it’s pinned, when moving content from any other site collection, you will see your pinned library on the Quick Access list.

Manage your pins from OneDrive.

Hope that helps!