2000 Chevrolet Lumina transmission

The most likely suspect is the radiator. Some cars have external oil coolers for the transmission fluid, but most have those coolers built into one tank of the radiator. Transmission fluid pressure while the engine is running can push the fluid through a leak into the coolant, which is bad. Cooling system pressure right after a hot engine is stopped can push coolant into the transmission cooler lines, which is real bad. Petroleum products in the coolant will rot rubber hoses from the inside, but that takes time. Water and antifreeze in the transmission don't lubricate moving and sealing parts like transmission fluid does.

If I'm right, you'll find an oily mud floating in the coolant reservoir.

Yup. Have the transmission flushed right away, and have the cooling system flushed soon. If you still have the red Dex-Cool, (Dex-mud) antifreeze, most mechanics will want to get that stuff out of there. Be sure to replace the coolant every two years, not every three years like it says on the stickers on your car, and don't consider it to be "lifetime" coolant like GM advertised. Water pump lubricant and corrosion inhibitor additives wear out in about two years, and acids form that corrode metal parts. This is real comm on GM cars but it usually causes leaks in the heater core first.

That shouldn't have gone into the engine oil. It will get sucked into the cylinders and burned. There will be real black exhaust smoke. That's not so serious because nothing gets sucked into the transmission fluid. This was pretty common on Fords too. Only Chrysler never used a vacuum modulator.