There's no officially prescribed way that I'm aware of, but this is a quick way to do it;
In these example, I have one object with a lot of loose parts that are all, themselves, manifold.
To modify a single object with multiple meshes
Enter edit mode on the object with Tab;
Change your pivot point to "Individual Origins";
Scale the meshes down to the ...
I think you may have edited the object after applying the RB, delete the the current RB sim and reapply with collision type as mesh.
edit:i think I know the problem your object does not move, normally rbs are added to moving objects by default, yours doesn't, causing some problems, i tried recreating the scene and ended up with messing in the sensitivity ...
I'm new to blender myself, but I think your solution would lie in the timeline key frames. For each object you can start movement at one key frame, move, and stop at another key frame. Hope this helps.
Based on the comment i would suggest the following solution:
Make a copy of the container object, adding a "-simhelper" at the end allows for easier identification later.
Select the copy and adjust the thickness of the solidify modifier to at least the size of the objects that are gonna fall in. Explanation a bit below.
Make the copy invisible in render by ...
Do you mean picture ...
Scene Properties > Riggid Body > Bake
Go to last frame of animation
Select all balls, go to camera view and switch to Edit mode
Unwrapp from View
Note: for animation I used Particles System (Plane object) Rendered as Objects (Sphere) - they are disabled from Outliner now. Particle system than converted to objects with addon Copy ...
The trick is simple: this is 'bake to the future'!
Once the simulation baked, go to the last frame and you'll know how to color the balls.
Here (but this is just an example as the materials could be totally different), I've assigned the object pass index knowing the positions at the last frame and used it to drive the color.
For Fridge (Passive) enable Animated. If Groceries didn't stay inside on Fridge move it can be because Fridge moves too fast, and some subsets would be needed for simulation - Scene Properties > Riggid Body World > Steps Per Second - increase value.
Note: Second gif looks like you have a hidden Cube with Passive Riggid Body in a Scene.