There’s a lot going on with the transmission, most of which you won’t see. However, I think the key to the problem is in clutches/bands. The clutches act just like the clutch pedal/throwout bearing/friction/clutch plate in a manual transmission. When you depress the clutch, the mechanical linkage between the clutch plate and the friction plate is broken. This enables you not only to change gears but also to synchronize he revs between the engine and transmission so that happens smoothly when you release the clutch pedal again. Instead of grinding, the takeup between gears is smooth.
Instead of having one clutch, the automatic has several, one for each set of gears. So, if your transmission is a five-speed auto then there are five clutch/band combos. They serve the same purpose as the clutch in the manual. They makes sure the engine and tranny are at the same rev point before they engage. Engagement is assisted by the bands. It’s usually not even noticeable, it is so smooth.
If, however, there is a clutch problem, then your car can buck and slip going between gears. This is exactly what you are describing. Getting this fixed does require a major transmission teardown and partial rebuild so you are probably looking at a $2,250 investment.
Be sure they swap out the clutch/band pairs together so that everything works and wears correctly.
While the transmission is open, have your technician take a close look at the planetary gearing to make sure they are engaging smoothly. This is the gearing that honestly changes when you go through the gears and it has to be working smoothly, as well.
If both need work, it’s probably a better idea to have a rebuilt tranny dropped in because it will save you money, even at $3,400.