Please take a look at this
post to see a couple of options.
Another approach that might be easier if you're going to keep all the windows in the same rows would be to set a solid color background on the topmost track (add one if needed), disable the existing background and all the other windows on the other rows, and produce that whole track.
Do the same thing for each line of windows, so you'll have 4 finished clips with one line of windows each. Make sure you're producing to the same profile that your finished video will use.
The only thing you'll need to do is to chroma-key out the solid background for each row, then use the PiP Designer as needed to shift the vertical location of each row. You can set opacity keyframes to fade rows in/out for the move, and placing any "foreground" clips lower on the timeline will allow them to show "on top of" the other clips as they fade in to view.
When you produce the final version, you might be able to use SVRT to quickly stitch all the clips and your desired background together.