As a workaround, I'm converting the Envelope geometry to a Polygon. This allows moving and editing the rectangles. However, the rectangles are loosing their right angles when editing them.
private void MyDrawObject_DrawComplete(object sender, ESRI.ArcGIS.Client.DrawEventArgs args)
{
    GraphicsLayer graphicsLayer = MyMap.Layers["MyGraphicsLayer"] as GraphicsLayer;
    Envelope envelope = (Envelope)args.Geometry;
    Polygon rectangle = new Polygon();
    PointCollection pointCollection = new PointCollection
    {
        new MapPoint(envelope.XMin, envelope.YMin),
        new MapPoint(envelope.XMin, envelope.YMax),
        new MapPoint(envelope.XMax, envelope.YMax),
        new MapPoint(envelope.XMax, envelope.YMin),
        new MapPoint(envelope.XMin, envelope.YMin)
    };
    rectangle.Rings.Add(pointCollection);
    ESRI.ArcGIS.Client.Graphic graphic = new ESRI.ArcGIS.Client.Graphic()
    {
        Geometry = rectangle,
        Symbol = _activeSymbol,
    };
    graphicsLayer.Graphics.Add(graphic);
}