I often need to take text, reshape it a bit from its original aspect ratio, then put it on a path.  The problem with Path -> Put on Path is that it restores the text to its original aspect ratio before putting it on the path.  Is there a way to do this that will leave the text shaped the way I want it?  Perhaps I can convert the reshaped text to individual paths then distribute these multiple objects along a path?

Yes, I'd use Path>Union to convert the text... it's a shortcut.

I know how to convert the text to one path per character, then union to make it a single path.  But, how do I distribute the text along a path after I have changed the aspect ratio of the text?

I'd try Path> Effect>Pattern along path 

That gets me closer to what I am looking for Tyler.  However, it distorts the letters.  See for example how the letter "H" gets wider at the top.  Any other ideas?

Arrange might be fiddly, but can be done (sloppy here):

I've used that one before.  What if the path is not a circle/ellipse/arc?

I would have suggested the group method of the Scatter extension, but it doesn't seem to be working. :-/

Thank you Tyler.  You have been very helpful.

Your question is quite intriguing, making me try to see what's possible without changing the text to a path. Hopefully the graphics below would help. The 3 ways are (1) the usual Put on Path, (2) Pattern Along Path with a copy-link to the text, and (3) Pattern Along Path with a copy-link to a clone of the text. In (1), you can change the text but not its ratio as you wanted. In (2), you can change the height and skew, but not the perspective or deformation e.g. making the top thinner like you wanted. With (3), you can add deformation or perspective to the clone and still edit the text. For (2) and (3), you can move the original off the page or to a hidden layer if you don't want it to appear in your design. Hope it helps.

Here is the Scatter extension used to place rectangle objects on the path, which are replaced by the text objects and spacer objects. This would be most helpful when there is a lot of text.

Note that the spacers and text are sequential in z-order.  Nice kerning would be done manually on the path.