I've been trying to figure out a similar issue and I'm so so thankful to the OP for getting this topic started. Thank you!
But, I'm dumb, and I have questions.

I tried this code, and I'm not sure exactly what I'm doing wrong here.
I noticed the code below this obviously wouldn't work on my machine as I have different folders.

    Set olFolder = Application.Session.PickFolder
    Set olDestFolder = Application.Session.GetDefaultFolder(olFolderInbox).folders("Test").folders("Test A"
I changed that section of the code to match my folders.
emailbox.jpg
So I thought the code should look like this:
Sub MoveMessages()Dim olFolder As Folder
Dim olDestFolder As Folder
Dim olItems As Outlook.Items
Dim strFind As String
Dim i As Long, j As Long: j = 0
    strFind = InputBox("Find what text?")
    If strFind = "" Then GoTo lbl_Exit
    Set olFolder = Application.Session.PickFolder
    Set olDestFolder = Application.Session.GetDefaultFolder(olFolderInbox).Folders("Inbox").Folders("Test")
    Set olItems = olFolder.Items
    For i = olItems.Count To 1 Step -1
        If TypeName(olItems(i)) = "MailItem" Then
            If InStr(1, olItems(i).Subject, strFind) > 0 Then
                olItems(i).Move olDestFolder
                j = j + 1
            End If
        End If
    Next i
    MsgBox j & " message items moved"
lbl_Exit:
    Set olItems = Nothing
    Set olFolder = Nothing
    Set olDestFolder = Nothing
    Exit Sub
End Sub
However, that gave me an object not found exception.
So what am I doing wrong here, I'm very new to Outlook VBA and a novice (though I have put together a few of my own programs for Excel and Word) in general with VBA.

I even tried just using this:
  Set olDestFolder = Application.Session.GetDefaultFolder(olFolderInbox)
That allowed me to actually bring up the search box but when I searched for "Emma" nothing moved.

Assuming I can get this figured out, is there a way to search message bodies for a series of string values?

Thanks!