I believe i understand that you have made!

I project curves Offseted
Trim the Volume
Kill curves
Select external faces of the "moved piece"
Inset 0 - Use separate height as you want (little measure of course)
Move in place
Boolean Union
It's a solid! :)

(it's a pity that we can't select directly only "Front faces" of the gap's face or i miss something :)



In fact we are obliged to move the piece even separated there is creation of vertical pieces at the angles!
That is not the case in the empty space! :) See animation above!