That got me thinking about how the Chamfer is produced. We can see just about the same narrowing were we to Pipe Cut the Path:

Essentially, a Circle Profile has been Swept along the Path. Self evident, the depth of Cut must vary in order for the Face width to stay the same; lacking variation in depth, Fillets help smooth those transitions, with limited blending of the Face widths. So, what to do? I considered a bunch of overly elaborate operations involving Planes Along Path, Projections, a bunch of Sketches- all of which I dispensed with by using Fusion's main strength- solid modelling. In the following illustration, I Chamfered all the lower Edges individually, then performed a Press Pull operation on individual Surfaces to bring their Edges into conformity, zooming in and using the digital control to adjust. 
Now THAT is what I expected from selecting all the Edges for Chamfering at once- but, through process, it's easy to understand why the tool performed otherwise. It would be cool if Chamfer featured a Conforming Edge selection option, eliminating the need to eyeball the Press Pull operation.