This seems to work -

Private Sub fcx_MouseDown(ByVal button As FLOWCHARTLibCtl.EMouseButton, ByVal keys As Integer, ByVal docX As Long, ByVal docY As Long)
    If Not keys = vbShiftMask Then Exit Sub

    Dim b As box
    Set b = fcx.GetBoxAt(docX, docY)
    If Not b Is Nothing Then
        Dim dataObj As MSForms.DataObject
        Set dataObj = New MSForms.DataObject
        dataObj.SetText b.UserString, vbCFText
        dataObj.StartDrag
    End If
End Sub 

OK in case anyone is interested, I solved this as follows:

I detected a mousemove with the button pressed on the flowchart control, saved the current box info into a variable during this mousemove event, then initiated an OLEDrag on the parent control containing the chart, and handled the OLEStartDrag event on the parent control.

If anyone has a better idea or thinks this approach may have problems, please let me know.

Thanks for your help.