I would have your objects implement interfaces like ICharacter, IEnemy, IHorizontalMover. The relevant design guideline is known as Favor Composition over Inheritance, and it should allow your design to be more flexible. One difference is that interfaces tend to specify behavior and any given object can implement as much or as little as it needs.

If you feel that this causes code duplication that you did not have when a base class just handled mostly non-overridden methods, then you should be using forwarding methods, where you implementing classes forward method calls to a private class (or stateless static class) that provides a function common to several implementations. This keeps the codebase DRY (meaning Don't Repeat Yourself). It can feel like you are writing more boilerplate lines to achieve what inheritance takes care of for you behind the scenes, but it is generally considered worth the effort, leaving the codebase more flexible and open to change than a pure inheritance design.

I would have your objects implement interfaces like ICharacter, IEnemy, IHorizontalMover. The relevant design guideline is known as Favor Composition over Inheritance, and it should allow your design to be more flexible. One difference is that interfaces tend to specify behavior and any given object can implement as much or as little as it needs.

If you feel that this causes code duplication that you did not have when a base class just handled mostly non-overridden methods, then you should be using forwarding methods, where your implementing classes forward method calls to a private class (or stateless static class) that provides a function common to several implementations. This keeps the codebase DRY (meaning Don't Repeat Yourself). It can feel like you are writing more boilerplate lines to achieve what inheritance takes care of for you behind the scenes, but it is generally considered worth the effort, leaving the codebase more flexible and open to change than a pure inheritance design.