Who would ever imagine that I will have to help with migrating SharePoint data, but that happened recently.
So there were SharePoint lists with lots of items and attachments which needed to be migrated to SharePoint Online.
This is how I did migration.
First I got Office 64-bit with Access, to process files larger than 2 GB. After that:
1) Open list, then select List - Open With Access
2) Wait for the list to be opened in Access, this could take really long if the list is huge. But you can see that Access file is growing, just to make sure that process hasn't stopped
3) Final step is to export data to SharePoint Online. So in Access choose External Data -> Export -> More -> SharePoint list, selec destination SharePoint site and list name and click OK
If the list is really huge, then process will be really slow and progress bar will stop, but again you can see in Windows Task Manager that there is action in networking thus making sure that process hasn't stopped.