Transmission is automatic. Here's what happens:

Start in 1st gear, get going,
change to 2nd gear, pick up speed,
change to 3rd gear,...
get to 4th gear if lucky.

If 4th gear is reached then everything's okay, at least for a minute. Then the trans fails and falls back to 3rd or 2nd gear.

If 4th gear is not reached the trans slips right away and stays in 3rd gear or falls back to 2nd.

Either way, after a downshift, the gear it was in is usually lost. And after a complete stop all the gears are lost and the car barely moves even with the gas pedal depressed, with severe slippage of the trans and jerking. The car does not move at all while idling in drive and reverse does not work.

Sounds like a mechanical problem, right? Well here's the weird part:

After the car is turned off and back on the transmission operates normally again with a repeat of the above.

I'm thinking of replacing the transmission with another (used) one, but that might not necessary.

Has anyone else had this happen?