АЛГОРИТМ

Настраиваем свойства элементов:

object Form1: TForm1
  DockSite = True
  DragKind = dkDock
  object Panel1: TPanel
    DockSite = True
    DragKind = dkDock
    DragMode = dmAutomatic
    OnDragDrop = Panel1DragDrop
    OnDragOver = Panel1DragOver
  end
  object Edit1: TEdit
    OnMouseDown = Edit1MouseDown
  end
  object Panel2: TPanel
    DockSite = True
    DragKind = dkDock
    DragMode = dmAutomatic
  end
end

И прописывем соответствующие действия:

procedure TForm1.Edit1MouseDown(Sender: TObject; Button: TMouseButton;
  Shift: TShiftState; X, Y: Integer);
begin
  if Button = mbLeft then
    Edit1.BeginDrag(false);
end;

procedure TForm1.Panel1DragOver(Sender, Source: TObject; X, Y: Integer;
  State: TDragState; var Accept: Boolean);
begin
  Accept := Source is TEdit;
end;

procedure TForm1.Panel1DragDrop(Sender, Source: TObject; X, Y: Integer);
begin
  Panel1.Caption := Edit1.Text;
end;