My car: 1985 Audi 5000 S non turbo automatic transmission 2.2 Lt.

The transmission won't engage into the next stage till the car has traveled a mile or so.
For example.. At 35 mph, the rpms read 2800. After the car has 'warmed up' .. gone a mile or so.. it functions properly.. 35 mph at 2200 rpms.
This ONLY occurs upon the initial start up.. After that , it's smooth transitioning as long as I don't let the car sit and cool down.
A year ago , it USED to take roughly 3 miles for this engaging process to kick in.. So it's getting better ( I hope)
I've changed the screen , filter, gasket and tranny fluid to the proper level.
Previous owner stated that it had a 'new' transmission put in. I've lost contact with the last owner so.. What do you think is going on here ?
Thanks much, Will