Okay, no problem.
PART 1
( Jeepers, this is going to be a long post, so I’ve broken it into separate replies) 
Here we go . . .
As you already know how to set up a rigid body system I’ll assume you have something similar to this where every mesh has the rigidbody physics on it:
Place all the rigidbody meshes that are going to ‘fall’ into their own collection and name this collection. [ it’s called “Bits” in my file ]. You need this actual name later on for the script.
I’ve used an Empty to drive the effect. You can use any object that has a location transform.
In essence what this effect is going to do is compare the Z location of each fracture bit to the Z location of the Empty and if the Empty is ABOVE ( ie higher in value ) then turn ON the rigid body’s Dynamic toggle causing it to start simulating:

By animating the vertical position of the Empty you can control when this happens.
Note. The centre of each fragment is marked by the dot that appears when you select a fragment, this was created when you ran the Make Fracture effect on your origianl mesh. It is the location of this dot that is used in the calculation for the fragment bit’s location. You can manually move these centres . . .
You may want to do this to control specific fragments to fall in a certain order to art direct the effect.
Now we need to add a driver from the Empty to each fragment. Pick ONE of the fragments, this will become the Active mesh and will be the one we create the driver on.
( There are 2 ways to add a driver, I’m showing the more indirect method for clarity in what it’s doing)
RIGHT click your mouse over this :

A dialogue box appears, pick Add Driver:
Another dialogue box will appear:
Ignore this, click anywhere and it will vanish. We now need to open the drivers window to edit it more easily. You can do this by creating another window panel or RIGHT Click again over the ‘Dynamic’ label and choose:
A floating panel appears and in here we fill in the drivers parameters. I’ve numbered them in the order to do them . . .
#1. Select this.
#2. From the drop down select Scripted expression
#3. click eye-dropper and select the Empty ( or whatever object you’re using to get a location ) in the viewport.
#4. Change this in the dropdown to Z Location. ( if you don’t see this option be sure to toggle here:
#5. Where it say “var” change this to “Empty_Z_Loc”. This is the name of the variable that holds the z value of the Empty, calculated on each frame of the animation.
#6. The Formula for the driver:
1 if ( Empty_Z_Loc >= Frag_Z_Loc ) else 0
copy and paste this line EXACTLY into the expression field. When you’re done it should look like this:
Now we need to copy this driver setup to all the other fragments. Select all the meshes then SHIFT-select the ACTIVE fragment last and using CTR-L link the Animation Data.
If you now pick any fragment it should show exactly the same driver in the driver window.
END OF PART 1.