Suppose I was making a Junkyard Special out of used truck parts and needed to have a 4wd transfer case as an auxiliary transmission.
if I had an automatic transmission, is it theoretically possible to shift the tranny into neutral, shift the transfer case from low to high, and shift the transmission back into drive, all while in motion?
Or can transfer case shifts happen ONLY with the vehicle fully stopped?
Depends on the transfer case. Some can be shifted from 2W to 4Hi on the fly, but have to stop to shift into 4L and I believe, but I’m not sure that there are some computer controlled ones that can shift on the fly to 4L
In the vast majority of light truck transfer cases the two gear ratios aren’t synchronized and since you can’t decouple the transmission and the transfer case there’s no easy equivalent to double-clutching a crashbox transmission. You might be able to do it by putting both the transmission and transfer case in neutral and then very precisely rev-matching when you put the transfer case back in gear, but it would probably be tricky and so time consuming as to not really qualify as shifting on the fly.
There are a few high end 4x4’s that have the transfer case ratios synchronized (like the Mercedes G-Wagens and I think some Land Rovers) as well as some specialty aftermarket off-road transfer cases, but the average one lying around your local junkyard isn’t going to be. Note that this isn’t to be confused with the more common “synchronized transfer cases” where the front and rear outputs are synchronized allowing for shifting from 2wd high to 4wd high even with the front axle disengaged.
I just want to say that Rick and Greasy are awesome! Giant Oak Kegs of Automotive Knowledge, and always willing to share.
Enola, what kind of hubs are you going to use? I can shift on the fly with my Jeep sometimes (pretty worn out), but can never run fast enough to turn the hubs! 
No actual project…just exploring the possibilities.
I can’t contribute much, other than to say that the chain drive NP231 in both Jeeps I had were easier to shift into and out of low range while the vehicle was rolling forward (with the trans out of gear).
Unless you need 4WD, why use a transfercase? There are ready made auxiliary transmissions availible. Look at old trucks, not pickups. By old I mean in the 1950s. Although some trucks still came with them into the 1960s. These are made to shift on the fly so no worries about screwing them up this way. You can still buy them new, look at the offroad trucking suppliers. Think mining and logging.
If you need 4WD, just get a divorced transfercase and install this auxiliary transmission between your main transmission and you transfercase. Problem solved. No need to reengineer something that already exists.
IHTH, 48.