Ok. So I'll do the cliff notes version.
Audi has updated the mechatronic on the DL-501 numerous times since it's introduction in 2010. They were a real problem in pre-2012 cars (just search the S4 section!). There was an update in late 2012 and again in 2014. The 2013's haven't really proven to be any less reliable than later model years despite there being an update. Solenoid part numbers are identical.
What DID change, as hahnmgh63 mentioned was the electronics, specifically the PCB board which is attached to the mechatronic. The PCB board update was a new encasement, new ATF-spec fluid and a larger heat sink. The fluid change was due to a reaction it had with the PCB board seal. The larger heat sink was a band-aid for not having a proper transmission cooler (enter the RMR transmission cooler!).
A physical rebuild of the mechatronic, solenoid replacement, is generally about $500 from specialty shops. Just for the rebuild. That does not include removal and installation. The PCB's tend to float at or below $500 but shop around.
Mechanically, the solenoids do wear over time. My theory it's due to viscosity breakdown of the ATF fluid not providing enough lubrication to the solenoid shaft in the bore. This is most likely due to heat (again, enter the RMR trans cooler!) and the breaking down of the ATF fluid.
I think a normal replacement cost for the mech, through Audi, is probably around $3K these days. That's removal and installation of a new/rebuilt unit.
It's rare though. You can do things to preserve the transmission. Change the fluids, both sides, every 20K. And install a proper transmission cooler (RMR as it's the ONLY one out there).
This is where service history comes in. The car will be far less likely to have issues with proper fluid change intervals. That goes for the transmission and rear differential. I wouldn't be consumed with it, just make sure it's been changed regularly.
Realize they use this same transmission in the A4/S4, A5/S5/RS5, A6/S6 and A7/S7 as well as the Q5/SQ5, Porsche Macan and a few others.
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