The answer is yes: having lived in 9 of the 50 US States in every cardinal (and sub) direction - as well as a stint in London - I’ve never encountered any confusion about this phrase. Limbs fall asleep. Or, a limb is asleep. But, if you’re sitting on your leg, and you know this to cut circulation, it would be unsurprising for you to say “it’s falling asleep” with the “degree” being the amount of paresthesia you expect to encounter when moving the limb.
I did search my memory before commenting, and came up blank: I have never heard about limbs falling asleep. Hence my initial question, which you hereby seem to have answered.
Apologies if you know this - I wasn't sure from your post.
My point was that exactly: The transitioning part sounds completely wrong to me. Where I come from, my leg can be sleeping or not, but it can't perform the state change from one to the other.
My native Danish is possibly fooling my perception. My leg can sleep ('sove'), but no way it could falde i søvn (the exact same construct as fall asleep).
The sleep part refers to the sensation of numbness. The tingling wake-up part seems to be associated with ants in large parts of the world. But not here. That would simply be tingling or prickling or some such thing.