I need a way to write a generic procedure to act upon an object type or any of its descendants.
My first attempt was to declare
procedure TotalDestro
In addition to what Lasse wrote, which is quite correct, most of the time you don't want to pass an object to a var parameter anyway.
An object is a reference type. What you see as the object is actually a reference to it. You would only want to pass an object reference to a var parameter if you wanted to change your object out for a new object. If you just want to be able to modify the members of the object, then you can do that by simply passing it to a normal parameter. Make method call take a TMyObject parameter instead of a var TMyObject parameter and it should work.
Of course, if you really are replacing the object, then feel free to disregard all this, and see Lasse's answer.