I have installed and added a new Exchange 2013 Server to my organization. I have moved mailboxes from the old to the new Exchange Server. I also wish to move the public folders and their content from the old server to the new server. They are both Exchange 2013 Server.

I have tried using this script New-MoveRequest -Identity "PF_SanFrancisco" -TargetDatabase MBX_DB01

but it returns this error

The operation couldn't be performed because object 'Betsafe' couldn't be found on 'xx.xx.xx'.
    + CategoryInfo          : NotSpecified: (:) [New-MoveRequest], ManagementObjectNotFoundException
    + FullyQualifiedErrorId : [Server=xx,RequestId=db2859ce-0811-4668-aba7-4a3dce87d296,TimeStamp=26-06-2015 1
   8:43:50] [FailureCategory=Cmdlet-ManagementObjectNotFoundException] 4AC0FD90,Microsoft.Exchange.Management.Recipie
  ntTasks.NewMoveRequest
    + PSComputerName        : xx.xx.xx

Any ideas?

You can try to export the data to PST files and then import. Or you may consider some other third party tool like, (http://www.exchangemigrationtool.com/) for PF migration.