Well, I ran some duct work from over the stove to the 1st floor. Then I installed a small computer fan to see if it would move air. It did. But I wanted something more permanent.
So, I purchased an in-line duct booster fan from airboosterfans.com. Most duct booster fans appear to have plastic parts and are only rated for 150 or so degrees. This one is rated for 275 degrees. (the 6" model is what I got).
Air Booster Fans.com
uct Fan-Inline Duct Fans-Garage Heaters-Electric Garage Heaters-Industrial Fans-Duct Reducers-Attic Exhaust Fans, Garage Heaters and More!
Not that I am going to let the duct work get that hot. I put a meat thermometer into the duct work before the fan and watch it. When the stove top is blazing away at about 500 degrees, the air temp in the duct work maxed out at 150. Most of the time, it is only around 100-105, though. Just right to keep the center of the 1st floor at 70-72 degrees. Perfect for us.
However, the fan runs all the time, and in the morning, as the fire dies down, the basement room that is is in cools off. Then the fan blows cool air. I don't want that. So I started looking around for a thermostat to control it. I found this one at Northern Tool.
Portable Thermostat | Heaters | Northern Tool + Equipment
It works great. It comes wired for heating, but it has instructions to switch it over to cooling by moving one wire inside on a terminal block. Really nice!

And I need to cool the basement room. Works great!
I set the thermostat for 75 and mounted it in between the floor joists about 8 feet out and above the stove. When the room temp up there hits 75 it turns on the duct booster fan and when the room drops below 75 it turns the fan off. No more cold air. Fantastic.
I recommend both of these products.