Engine compression versus braking downhill

We took my brother-in-law with us camping this weekend. I down shifted out of overdrive into 3rd in my automatic when descending any grade more than a mile long. My brother-in-law, who, in fact, knows a lot more about cars than I do, suggested that I should not do that. He said it was hard on the valves and it is cheaper to replace brakes than valves.

What, if any, damage or premature wear can be caused by using engine compression instead of the brakes?

Perhaps I should only use compression on either very steep descents (where 2nd gear would be a better choice than 3rd) or for very long descents (more than 2 miles long).