This is a necro-thread from 2020. But since it "tis" the season for stuff like this to come up, and I'm bored, why not?
I'll add my vote to moving cool air to the stove, not vice-versa. It seems to work better for whatever reason, and it's easier to have a fan on the floor anyway. Maybe it's because the floor is the part of the house that is unencumbered by obstructions. On average, doorways don't extend to the ceiling. I believe that energy loss in fan propelled air is going to be subject to the laws of affinity. Fan pushed air moving rapidly hitting an obstruction is going to lose energy at a square of its velocity, so much (4x) better to have the slower-moving return air working its way around the obstructions.
Once it comes to moving air around in ducts, I doubt it really matters as long as all things are made equal by having all the ducting within the insulated areas of the home. Physics kinda dictates that if you have X amount of air, and Y amount of heat, how you swish that around isn't going to change the total. If you move more of it to the colder part of your house, or more of it from the colder part of your house, you're not going to change the total, but the more you move, the warmer the outskirts of your house will be (and the cooler the center).
I think most of us heating with a wood forced air furnace are familiar with the concept. I know for me, the trick is to move as much relatively lukewarm air as possible to the outer edges of my house. During the winter, that means that my wood furnace blower runs 24/7 pumping air through flow-enhanced outlets at the outskirts. All central vents are closed. The returns are all located in the central parts, but despite the cold air moving through there, the central part of my house always feels a little warmer. I think this is because of the radiational effect - there are no cold outside walls/floors/ceilings to absorb heat, so the warm surrounds make it feel warmer. Of course, that's only necessary because most of us are using a HVAC duct system designed for an on-off traditional fuel furnace that has a much larger blower and a much warmer output. We've traded the house-wide on/off temp swings for the temp variations room-room. Pumping the only lukewarm air, only to the distant rooms, compensates for the larger heat loss there.
Back on topic: I think if you could move ALL of the air through the room that the heat source was in, using your HVAC to even out the temps would work (physics X, Y, yada, yada). But, if you're only moving a fraction of the air through that room, as any normal HVAC system would be designed to do, the results are going to be disappointing. They disappoint me. If I circulate mass quantities of air, by leaving it on all the time, it helps-some. This is offset by the cooling effect on humans by the air movement. If your ducts can lose heat to the outside, it could actually hurt. Having tried both, using floor fans at least moves all the air I'm actively trying to heat through the heat source room, and works better because the target room with the first fan in it gets the benefit, and usually the other rooms that I'm not motivated enough to put a fan in, are getting good enough natural circulation.
Real world, if you're trying to keep the plumbing in distant rooms from freezing, and all you have is a central wood stove and a HVAC system, turn the blower on. It'll help because the temp delta is high enough, and your plumbing doesn't feel drafts. I've been there. But, if you're trying to be comfortable, it probably won't help much, or it may make you less comfortable. Try it, but don't plan on it as a replacement for your existing HVAC heater.