I have had the same problem on my 2013 Cayenne S (under CPO ~ 66,000):
Symptoms: low RPM and hard pull hesitation, lapse in power, "thumping," etc. as described by others. Initially the problem only occurred sporadically but became an every drive occurrence and then, near the end, a cause for significant alarm while operating. Just before the fix was installed, I could not rely on the car to accelerate when needed.
PROBLEM: Bad Transfer Case, most likely due to over-heating due to an initial failed design.
Explanation: With the transfer case out, the