It will sorta do that too as that air stays in the house, it just gets moved out the room. The heat will get pushed where the wood stove is, and that air will eventually get sucked into the furnace return.
I have a few ideas for the wood stove too. Will add a 6" duct above (that part is already done, it's just not terminated anywhere yet) which will either go to the furnace return, or dump into the crawlspace or maybe just dump directly on the lower floor (above crawlspace), which is coldest part of house. If I opt to dump it in the crawlspace that obviously means the server room air will be hot, but I will add a water cooling loop with a radiator at the intake and one in the garage. That way I can heat the garage too. I will probably do that anyway at some point since I do want heat in the garage once I insulate it. Just really want it like 5-10 degrees so it's comfortable enough to work in. May as well use free heat for it. To create a proper air cycle it seems dumping it straight to the crawlspace makes the most sense though. I would add some floor grates in appropriate areas so some of the air flows up, since the 6" duct will be moving a lot more air than the 4" one and I planned it that way.
For now I'll just focus on the server room itself though. At this rate I don't even know if I'll get my wood stove going this year. I really should have just done the chimney myself in the summer, but any day off I had where it was nice I was doing work on my off grid land instead, or yard work at my house that had to also be done. Only get so many days in a summer to do that stuff. I kept getting the run around from my contractor and next thing you know we're in middle of winter again. I found another that says he will do it but that was like a month ago. I still have work to do on the inside so been focusing on that. Once I'm 100% done inside then I will just do the chimney myself even if it means doing it in the cold. It's not really hard it's just tedious, since it involves going through the soffit and the roof. I was just being lazy by hiring it out. The inside part also spawned a lot of misc projects, such as rerouting plumbing, electrical, drywalling inside the server room since I can't get the sheets in if I only drywall the outside and plan to do it later. Come to think of it I did quite a lot of work down there even though it feels like it's dragging on. Only get so many days off, even with my shift work. I would really hate to go back to a 8x5 job since I'd have even less time.