Hi there !
There are multiple solutions to this problem and design is depends on the end feeling you want to achieve.
1- You can just attach object to cart. After placing the object you can detect and attach to cart socket or actor itself so they maintain position. Also you can use physics constrains to maintain a wobble motion still in the cart even.
2- When objects are inside the cart their physics properties changed, like weight, friction also damping. Damping would be quite important in that terms.
3- You can use mix of match of methods, an experimental method would be (depending on the game) use an invisible physics field on top section of the cart that pushes physics objects back into cart. A good phsyics material change and field addition on cart velocity > 0 would achieve quite nice results while still maintaining phsyics inside the cart.
I am assuming you are not using movement on physics so it shouldn’t add unwanted physics bugs.