Public Folders migration to Office 365 and on-premises Exchange is not an easy task with manual methods for this scenario professional tool is a must and the Shoviv
Exchange Migration Tool is appropriate for this. It allows smooth migration of multiple numbers of source mailboxes like EDB files, Outlook Stores, Office 365 mailboxes, Live Exchange mailboxes, etc. into Live Exchange (On-premises), Office 365 and Outlook PST files. Along with this, Mailbox mapping option, Search Option, Filter Option and option to preview added source mailbox are also given. It is capable to smoothly migrate Live Exchange/Office 365 Public folders and Archive Folders into another Live Exchange or Office 365 Public and Archive Folders. To check its working a free demo/trial version of the software is also available.

For more information -
Public Folder Migration