I don't know of a way to do it before it has been displayed, but once it is on the screen you could use an approach like this:
private Image GetControlThumb(Control control, int thumbSize)
{
    Bitmap imgLarge = new Bitmap(control.Bounds.Width, control.Bounds.Height);
    using (Graphics g = Graphics.FromImage(imgLarge))
    {
        g.CopyFromScreen(
            control.Parent.PointToScreen(new Point(control.Left, control.Top)),
            new Point(0, 0),
            new Size(control.Bounds.Width, control.Bounds.Height));
    }
    Size size;
    if (control.Width > control.Height)
    {
        size = new Size(thumbSize, (int)(thumbSize * (float)control.Height / (float)control.Width));
    }
    else
    {
        size = new Size((int)(thumbSize * (float)control.Width / (float)control.Height), thumbSize);
    }
    Image imgSmall = imgLarge.GetThumbnailImage(size.Width, size.Height, new Image.GetThumbnailImageAbort(delegate { return false; }), IntPtr.Zero);
    imgLarge.Dispose();
    return imgSmall;
}
You can use it to get a thumbnail of any control, like this:
myPictureBox.Image = GetControlThumb(someControl, 100);