Verbals such as "Someone's doing something" or "Someone doing something" are common. But when it comes to verbals in long structures, it gets confusing.
Here are some sample sentences with such structures.
1-She was against French invasion of England.
2-She was against France's invasion of England.
3-She was against France's invading England. (as in the case of "I don't mind his living with us.")
4-She was against France invading England. (as in the case of "I don't mind him living with us.")
In my opinion, they all seem to work grammatically. However, I am not sure whether they are all correct or interchangeable?