
Currently, I have the "lots of code" as noted in the diagram inside each child's constructor. My goal is to move it to the parent's constructor.

Currently, I have the "lots of code" as noted in the diagram inside each child's constructor. My goal is to move it to the parent's constructor.
Is it possible to call a virtual method from an abstract class constructor?
Technically it is possible, but it won't work as you expect, so don't do it, because the virtual table for derived classes has not been constructed yet.
The implementation of the class being constructed is going to be called when a virtual function is invoked from a constructor, and if the virtual function you are calling is pure, you get undefined behavior.
Per paragraph 10.4/6 of the C++11 Standard:
Member functions can be called from a constructor (or destructor) of an abstract class; the effect of making a virtual call (10.3) to a pure virtual function directly or indirectly for the object being created (or destroyed) from such a constructor (or destructor) is undefined.
If you try to invoke initializeFiles() from the FileContainer constructor it will invoke FileContainer::initializeFiles(). This is because the constructor for the derived class has not been executed yet and therefore the v-table for the derived class has not been built.
Also if FileContainer::initializeFiles() is a pure-virtual function then you will get a crash.