How to prevent an abstract class with public derived classes from being inherited in other assemblies?

To make InternalData internal, DoProcess must be private or internal (or InternalAndProtected, but C# doesn't support this CLR feature). It can't be protected or protected internal.

internal abstract DoProcess(InternalData internalData);

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I'd probably also add an internal abstract void DoNotInheritFromThisClassInAnOutsideAssembly() member. That prevents anybody outside the assembly from inheriting from your class, because they can't implement that member and they get a reasonable compiler error. But you can't make the Base class itself internal.

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I'd consider refactoring the code, so that you have no common base class. Probably by using some internal interfaces and composition.