I'm new to Inkscape, but this does not seem obvious from the docs and pdfs that I've seen so far:

Some objects can simply be dragged via mouse (w left button). The same objects can often be selected with a single click.

Other objects cannot be dragged. They need to be selected, and then keyboard cursor keys used to move them.

I've tried a couple tests, like attempting to convert the objects or break into components, but that does not appear to be a solution.

Is there some attribute that causes this? I would have thought that any object that was movable via cursor keys should be movable with the mouse.

The cursor has to be over a nontransparent part of the object
e.g. if an object has a stroke but no fill the cursor has to click right on the stroke to drag the object.

I just tried that, and it works. Thanks, Druban.