C# has abstract classes and interfaces, should it also have “mixins”?

The problem you describe could be solved using the Visitor pattern (everything can be solved using the Visitor pattern, so beware! )

The visitor pattern lets you move the implementation logic towards a new class. That way you do not need a base class, and a visitor works extremely well over different inheritance trees.

To sum up:

1. New functionality does not need to be added to all different types
2. The call to the visitor can be pulled up to the root of each class hierarchy

For a reference, see the Visitor pattern