Airlayering rootstock

Hey y'all,
If you airlayer a fruit tree that's been grafted onto dwarf or semi dwarf rootstock, will the roots the airlayered branch develops be full sized (not dwarf) and will the resulting tree be full sized and unusably vigorous?  

As I understand these things, your rooted cutting is a genetic copy of the top section of the grafted tree.
So it will have all the characteristics  of the 'full sized' variety that was grafted onto the top. including full sized roots.

I you want to 'duplicate the grafted tree that you have, you would need to air layer a 'sucker' growing from below the graft (the root section)  and graft a cutting from the top portion of the tree onto it. This would duplicate the characteristics of of your original tree.

please forgive my ignorance here,  why would someone graft dwarf rootstock to a full sized tree?
I do not see what benefit you would get: My understanding is that grafting is done to take a variety with a desireable fruit (and less hardy roots) and give it the advantage of 'good' roots from a variety that grows well but doesn't have nice fruit.
Other than stunted rootstock that might fail to keep up growthwise with the top (unless heavily pruned?)

If you root a sucker taken from below the graft you will get a copy of the rootstock and it will retain the characteristics of the original rootstock, including its dwarfing  characteristics. Same thing applies to airlayering.

And if you root or graft a cutting (scion) off  the tree above the graft you will get a full size copy of the desired apple variety. If you airlayer a branch from above the graft on your original tree the resulting growth will retain all the characteristics of the variety and the rootstock will have no bearing on that resulting tree. If it did retain the vigor of your variety and the rootstock we could forget about grafting.

To get a copy of your current tree including its dwarfing characteristics you would need a rooted sucker from below the graft and then graft a cutting from above the graft on the original tree.