Call a common named method of all base classes from a derived class with multiple inheritance

I realize that what I am trying to do is possibly wrong, so I would gladly accept and alternative approach. All code below is just an illustration.

Suppose I have a derived class, with multiple base classes:

template<typename... Bases> class Derived : public Bases...{
public:
    VariadicTemplate(BaseClasses&&... base_classes) : BaseClasses(base_classes)... {}

    void DoAllFoo();
};

And all the base classes have a method, named DoFoo(). I was going to derive them from a common base with a public method Foo():

class CommonBase {
public:
    virtual Foo() { DoFoo() };
protected:
    virtual void DoFoo() = 0;
};

class Base1 : public CommonBase {
    DoFoo() { /* displays something */ };
};

class Base2 : public CommonBase {
    DoFoo() { /* displays something else */ };
};

....

Now this is how I was going to use it. I want to instantiate some amount the Derived class objects, specified with different Base classes:

Devired<Base1, Base2> x;
Devired<Base5, Base7, Base21> y;
...
x.DoAllFoo();
y.DoAllFoo();

I want (somehow) the Derived's DoAllFoo() to "iterate" over it's Base classes and call each classes Foo() method.

The only solution I can imagine is to have some set of function pointers/functors/whatever in Derived and make all Bases register their Foo() method in this set when constructing. This will probably work, but look like it could be done better.

I hope, that I am not familiar with some common pattern to solve this problem (or I am just pure wrong), so please advise. Thank you.