Hello and welcome to JustAnswer!
What year, model, and engine do you have?
This is not an uncommon problem on the later model 41TE transmissions with the NGC engine/transmission controller. A loss of prime problem causes this. You may have a problem with the pump in the trans, a leaking low/reverse piston in the trans, or most likely-low fluid. You may notice when you start the car cold it hesitates a little before it moves if you don't warm the car up. Here's what happens.
When you start the engine the low/reverse clutch automatically applies. When you put it into drive it applies the underdrive clutch. If the trans has lost prime the fluid has to displace the air in the clutch circuit. There is a pressure switch so the trans controller knows that the clutch is applied-it watches for fluid pressure in the circuit. It monitors the time between you putting it in gear until it sees the pressure switch close.
Your trans is fully adaptive-it monitors every shift (each shift is actually 3) to make sure its applying the correct amount of fluid with the solenoids. If it shifts hard, it lowers the clutch volume index (cvi), lowering the cvi makes it apply the clutch with less fluid-less force. If it shifts soft, it raises the cvi. When the loss of prime happens on a cold start, it actually counts up the cvi for the underdrive clutch while the pressure switch is not closing because it thinks it hasn't applied enough fluid.
The underdrive clutch comes on and stays on for first, second, third gear. For fourth gear they turn it off and add the 2/4 clutch. the only shift you feel that is actually solely the underdrive clutch is the 4-3 downshift. After the cvi's climb so high on a cold start, it remembers that high cvi, and on every 4-3 downshift it slams the clutch on hard, giving you this jerk you feel.
The first thing you need to do is get the trans hot and check the fluid level, I'm betting its low. Internal problems in this trans are getting pretty slim these days. I'll bet you either have cooler lines or a solenoid pack on the front of the trans leaking and its caused this whole thing. Top the fluid off using only Chrysler ATF+4. To get rid of the harsh downshift you'll have to find someone with a transmission capable scan tool to do a quicklearn procedure. This sets all the cvis back to a baseline to let them readapt and you will be good to go. This can be done by a dealer in just a few minutes.
If you have anymore questions let me know, this may have been a bit confusing.