How to detach a panel and show it in a separate window?

As noted by others, there are several problems with changing the parent window of a control without changing the ownership, and changing a controls owner can be difficult if it has multiple controls sitting on it...

One way around it is to use a frame instead. A frame owns all of it's sub-controls, so all you would need to do is change the owner and parent of the frame, and everything else will come along with it. This approach also allows you to keep all the event handlers and glue code in a single place as well.

N@

You have to take ownership into account, otherwise the destruction of form A would lead to the disappearance (i.e. destruction) of your panel on form B, or worse.

type
  TForm2 = class(TForm)
  public
    InsertedPanel: TControl;  // or TPanel 

.

procedure RemoveComponents(AForm: TComponent; AControl: TWinControl);
var
  I: Integer;
begin
  for I := 0 to AControl.ControlCount - 1 do
  begin
    if AControl.Controls[I] is TWinControl then
      RemoveComponents(AForm, TWinControl(AControl.Controls[I]));
    if AControl.Controls[I].Owner = AForm then
      AForm.RemoveComponent(AControl.Controls[I]);
  end;
  AForm.RemoveComponent(AControl);
end;

procedure TForm1.Button3Click(Sender: TObject);
begin
  Form2.InsertedPanel := Panel1;
  Panel1.Parent := nil;
  RemoveComponents(Self, Panel1);
  Form2.InsertComponent(Form2.InsertedPanel); // < this is not necessary
  Form2.InsertedPanel.Parent := Form2;        //   as long as Parent is set
  Panel1 := nil;                              //   or if you free the panel
end;                                          //   manually

The extra reference may seem a bit silly: Form2.InsertedPanel and Panel1 point to the same object, but it's kind of semantically preferred. Maybe a central controlled variable is better.

Update:

I falsely assumed that RemoveComponent cascaded to the child controls on the panel. It doesn't, of course, so only removing the panel from form A would leave all the child controls of the panel still owned by form A. So I added the RemoveComponents routine to remove ownership from all the child controls of the panel.

Note that the child controls of the panel don't have an owner at this time. But since they are parented controls of the panel, destruction of the panel will free those controls. So be sure the panel has a parent, or free the panel explicitly.

All of the above only applies to a design time created panel, placed design time on the form, which was my assumption. Since this changing parents behaviour is apparently wanted or needed, you might want to consider to implement it completely at runtime. To keep the abbility to design the panel designtime, I suggest to create a Frame on which you can design that panel, and jump the Frame around your forms.