This is by design. The MouseDown event captures the mouse, Control.Capture property. The built-in MouseUp event handler checks if the mouse is still captured and the mouse hasn't moved too far, then fires the Click event. Trouble is that calling DoDragDrop() will cancel mouse capture. Necessarily so since mouse events are now used to implement the drag+drop operation. So you'll never get the Click nor the DoubleClick event.
Controls that both need to respond to clicks and drag+drop are a usability problem. It is fixable however, what you need to do is ensure that the user has moved the mouse enough from the original mouse down location, then start the drag. Make your code look like this:
Private MouseDownPos As Point
Private Sub Dir1_MouseDown(ByVal sender As System.Object, ByVal e As System.Windows.Forms.MouseEventArgs) Handles Dir1.MouseDown
MouseDownPos = e.Location
End Sub
Private Sub Dir1_MouseMove(ByVal sender As System.Object, ByVal e As System.Windows.Forms.MouseEventArgs) Handles Dir1.MouseMove
If e.Button And MouseButtons.Left = MouseButtons.Left Then
Dim dx = e.X - MouseDownPos.X
Dim dy = e.Y - MouseDownPos.Y
If Math.Abs(dx) >= SystemInformation.DoubleClickSize.Width OrElse _
Math.Abs(dy) >= SystemInformation.DoubleClickSize.Height Then
'' Start the drag here
''...
End If
End If
End Sub
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