Interaction between engine and transmission cooling?

My '16 V6 AT is at the dealership with a bad shuddering between the 2-3 and 3-4 shifts at low acceleration. The dealer is telling me that engine coolant is leaking into the transmission through the transmission cooler, causing the shuddering, and since the cooling system is not part of the powertrain, the new transmission and cooling system won't be covered under the extended warranty. Because of course it isn't.

The best I can tell, there is no interaction between the engine coolant and the transmission cooler. It seems to me that the transmission cooler bolts to the front of the radiator, but only transmission fluid flows through the cooler.

Can anybody explain how engine coolant can leak into the transmission?

 

Transmission Fluid also flows through the radiator, which has a small tank separate from the coolant for transmission fluid. This is how AT transmissions are cooled when they don't also have an additional external cooler plumbed inline with the radiator.

There must be a leak somewhere in the radiator or along the way allowing the two to mix.

 

Trans fluid and coolant interact in two locations, the warmer attached to the side of the transmission, and the cooler inside the driver side coolant radiator tank.

See diagram here https://www.tacomaworld.com/threads/missing-transmission-cooler.671079/page-14#post-23586135

 

Awesome, thank you for the info.