Looking for information on transonic shooting. Specifically, if you do not have DOPE into the transonic flight stage. I do not believe in the supersonic/transonic shift, as in it makes the round tumble, but what effect does it have exactly?

Are you really going to be shooting through the transonic range?  Cause that is pretty out there for most shooters.

I highly doubt that bullets "tumble" when they drop through the sound barrier but I do suspect they receive some buffeting and possible yaw effects.  There are a ton of other effects at very long range that can effect accuracy.  Its not at all like shooting at regular distances.

From experimenting with subsonic rounds I can tell you that even a bullet that wildly tumbles and keyholes can still be fairly accurate.  At velocities below speed of sound, the BC goes way up and a tumbling bullet doesn't seem to fly off erratically as we might assume.  I've seen some reasonable groups where bullets were going through the paper completely sideways.

Pertinent to this discussion is high velocity 22 rimfire.  They start off at ca 1200 fps and drop back though the sonic barrier fairly quickly.  If you look at all the "match grade" ammo, it is all regular velocity to avoid the trip back across the sound barrier.  This tells us that the transonic range is not conducive to the best accuracy.

It could happen in some extreme cases with ridiculous angles of repose but it'd require something of a pumpkin slinger of a cartridge to begin with and a really long boat tail and nose ogive and dense as hell air. If the bullet was ever stable then it's going to get more unlikely that it'll destabilize as it slows down, I'll just leave it at that.

As far as what happens... Pretty much ^^^. What you'll see in the real world is you're banging away at (distance chosen arbitrarily for the purposes of illustration) 700yrds hitting consistently and 725yrds seems like it's no different. By 750 you're seeing your dispersion grow with a surprising number of misses and by 800 any hits on the target are nearly accidental with impacts all around the target but not on it and certainly never 2 in a row. I have seen it too many times to want to think about. The most blatant was a guy launching NATO spec .308 147gr FMJ's over empty terrain. Up to 700 we were on like Juan  with first round hits every time. Then at 800 he couldn't land a round to save a life and looked back and said, "Do you think they're destablizing at the transonic?" which was answered with, "Yes. They definintely are." which was easy enough given what was being seen. Put him on a different load spec and there's nothing inside 1000yrds he couldn't first round.

When someone comes for the first time to the 1K range and tries to shoot something that won't stay supersonic at that distance (often a .30 caliber with the wrong bullet) we see great degradation of accuracy.  In the pit, you know immediately if a round has gone sub-sonic on its journey to the target.  When those go overhead, they end up all over the paper and off the target completely.  Often they hit the berm, pissing everyone off in the pits.

Short answer, the bullets start to wobble, and once that happens they can go anywhere -- but, usually not where you want them to.

^^^ this. Get the right tool for the job. Go bigger, longer, faster than you need and you’ll never come up short.

Just ask my wife. Or actually, please don’t.