Flux Transfer of Current also involved in Autotransformer?
Do this thought experiment: take a normal transformer.  Connect the non-dot wire of the secondary to the dot wire of the primary.  You now have an autotransformer.

So yes, the mechanism is the same.

I suspect that the reason your flux readings don't change is because you were not reading the flux inside the core.  Even if you were able to do so, in a reasonably efficient transformer that's operating even remotely normally* the magnetizing flux is going to remain substantially the same.

* "remotely normally" in this context means it's not bursting into flames.