Assuming the items are in the inbox there is this:
New-MailboxExportRequest -Mailbox MyMailbox -IncludeFolders "#Inbox#" -ContentFilter {(Received -ge '02/23/2005 00:06:00') -and (Received -lt '02/24/2005 12:00:00')} -FilePath "\\server\C$\PST-Exports\exported.pst"
That one above won't remove the items from the mailbox, it just dumps a copy out into a PST.
I have had trouble with this one below trying to get it down to the minute but you can get a report sent to another mailbox with this:
Search-Mailbox -Identity MyMailbox -TargetMailbox SomeMailbox -TargetFolder test -SearchQuery '(Received:02/23/2005..02/24/2005)' -LogOnly -LogLevel Full
That search-mailbox one goes by UTC time for some reason so it misses some hours based on how far off from UTC you are and as I mentioned I haven't been able to get it to work entering the time, just the date.
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Edited by
MnM Show
12 hours 41 minutes ago